ti 3ilTfrTT
li 3l TMTAGE
The Bulletin of the International Linear Algebra Society
Seruingthe lntemationalLinearAlgebraC-ommunity
Issue Number 20,PP.l-32, APril L998.
Editorial Assistanfs:Slrane T. |ensen & Aline S. Normoyle
tel.(1-514)398-3333
and Statistics
of Mathematics
Department
(1-514)3e8-3899
FAX
1005
Room
Hall
Burnside
McGillUniversity,
StreetWest
805ouest,rueSherbrooke
rtyaneMath ' McGi 11 . CA
Montr6al,
Qu6bec,CanadaH3A2Ko
Editor-in-Chief:C'eoqgeP. FI. Sfi AI{
(1989-19!8);
Hited by: RotrcrtC. THOMPSON(l%S-l!49); StevenJ. I".EON& RobertC. THOMPSoN
stevenJ. LEON (lry3-lgg4); StevenJ. LEON & GeorgeP.H. STYAN(199+-199,7)
Editors: Chi-Kwongu, SimoPUNTANEN,HansJoachimwERNER
senior.Associate
AssociateMitors.' S. W. DRURY,StephenJ. KIRKI"A,ND, PeterSnn4ru-' FuzhenZHANG'
Prize(EmilioS@icato)
""""""'2
International
EgervAry-ABS
......'................"11
Weave|
(James
R.
1998
28,
1,
l997-February
March
Report:
tUnSfreasurer's
Book Feviews
R. B. Bapat& T. E. S. Raghavan Nonnqative Mauies aN Apfliations (FuzhenZharg) """""'2
""""""""""'2
Ali S. Hadi:MatixNgefuaas a Tool(1.S. Alaloufl.....'..'......
""""""""""'3
RusselfMenis:MuttilinearNgaMa(S.W.Drury)
FuzhenZhang: LinearNgefia-Chaltengirp Problemsbr Students(WasinSo) .'........"""""""""4
New&ForttrcomingBooksonLinearAlgebra&RelatedTopics:1997-1998 """""""'4
Linear Algebn EYenfs
Portugal.....
Maylt-August 7, 1998:Coimbra,
lvtaytS, 1SSS'ttongKong..........
'..............
Wisconsin
June3-6, 1998:Madison,
Ergland
June 16-17,1998:Marrchester,
Jufy 3O-.31,1998:Victoria,BrltishColumbia
FlorkJa
December11-14,1998:FortLauderdale,
fmage Problem Corner
Matrices
t&t: SiS ComplexHadamard
1&4: Boundsfor a Ratioof Marix Traces
1$1: E(lenvalues
DefiniteMatrices
1$2: Non-negative
witha Sumand Product......
Associated
193: Characterizations
DefiniteMatrices
194: Eigenvaluesof Non-negative
................
Definiteness?
Product
15: Symmetrized
NewProb1ems...,.............
printedby the McGiiluniversityPrintingservice. Printedin canada
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poge 2
lmage; No. 20
Egervory-ABs
InternotionolPrize
With referenceto the intensivedevelopmentof ABS metbodssince l98l at the University of Boguno and to the seminal,albeit
Iittle-known, worls of tbe late ProfessorEgentfiry,wherethe classof algorithmslater called the unscaledABS classwas already
developed,an internationalprize is now offeredby the Departmentof Mathematicsof the University of Berganrofor the bestwork
in theABS neK writtenby a singleauthorwhoseageon December31,lgg7,was no morethan39 years.The prizeconsistsof two
million Italian lira" plus travel expenses(economyclassif by air) andliving expensesfor threedays,to cometo Bogamo to presentin
a seminarthecontentof the winning paper.Submittedpaperscanbe eitherin preprint or alreadypublishedform. They mustbe sent
to: TheDirector,Departmentof Mathematics,
Universityof Bergamo,p.zaRosate2,I-24129Bergamo,Italy. Age mustbecertified
by a copyof thepassport.Papersmustbe receivedby June30, 199E.The winnerwil be deciclect
by aninternationalcommissionof
expertsin ABS methods.The winner will receivetheprize in Bergamo,wherethe work will be presented.
Eutt-to SpEDlcATo:Director
Depanment of Mathematics, University of Bergamo
p.za Rosate2, I-24129 Bergarno,Italy
FAX +39 35 249598,emilioGibsuniv. unibs . ir
lBooK
reaiurus
R. B. Bopot & T.E. S. Roghovon: Nonnegative
MatricesandApplications
Reviewedby FuzhenZhong,NovoSoutheostern
FortLouderdole
University,
NowrcgativeMatrices and Applicationsby R. B. BapatandT. E. S. Raghavan[Encyclopediaof MathematicsandIts Applications,
Vol. 64, CambridgeUniversityPress,xiii + 336pp., ISBN 0-521-57167-7,19971
is a very goodtexr on ma6ix theoryfor graduare
studentsand a valuablereferencebook for linear algebraistsand researchers.The book containssevenchapters,beginningwith
the Penon-Frobeniustheory and maEix games,andthencoveringdoubly stochasticmatrices,inequalitiesfor positive semidefinite
matrices,topicsin combinatorialtheory,andapplications
to statisticsandeconomics.Thebook presentsthe theoryof nonnegative
ma8icesin an elegantway and showsconnections
of thesubjectwith gametheory,combinatorics,
optimization,andmathematical
economics.The book is basedon the authors'researchofrecentyearsin the field, thoughsometopicsarestandard.Thefirst five
chapterscontributeto a good text on basicnonnegativematrix theory for first-yeargraduatestudents,with an option of Chapter6 or
7' if teachingtime permits. Someexercisesaregiven after eachchapterbut studentsmay expectmore. The pierequisiteis a good
elementarylinearalgebracoruseandsomebasicmatrixskills.
Ali S, Hodi: MatrixAlgebroos a Tool
Reviewed by l. S.Alolouf, Universit6du Qu6bec d Montr6ol
In the author'swordsthemainpurposeof theMatrixAlgebraasaToolby Ali s. Hadi [Duxbury,xi+212pp., ISBN 0-534-237124,
I 9961is to presentmatrixalgebraasa tool for studentsandresearchers
in variousfieldswho find it necessary
for theirown [statistical]
work. This is no easytask,sincepractitionersneedto be acquainted
with sophisticated
notionssuchas vectorspaces,eigenvalues
andeigenvectors,
andmatrix diagonalization,
andtheyneedto reachsuchlevelsfrom relativelymodestmathematical
beginnings.
Thebook,however,servesexactlythat function: startingfrom thedefinitionof a matrix andmatrix operations,it runsthroughthe
fundamental
notionsin barelymorethan200 pages.For peoplewith limited mathematical
background,this is themostdirectroute
to autonomyin linearalgebra.Of course,thereis no room for proofs.Theoremsarestated,andthey aremadecrediblethroughnumericalexamplesandcounterexamples.
Theseexamplesaresurprisinglyeffective,andfor thosenot mathematically
inclined,they
aremoreconvincing,andtheyoffer moreinsightthanformalproofswould. (Proofsarenot entirelyabsent,bowever:theoccasional
poge 3
April 1998
practitioners,which meansthat
corollary is proven,usually to pedagogicaladvantage.)The subjectmatteris gearedspecificallyto
areno minimal polynomials'
manyof the thingsthat arsstan-dardfare in introduciorybooksareeitherglossedover or omitted: there
areexpressedin termsof
no elementarydivisors,no dual sp,rces,no Jordanronns. Ano the approachis coordinatebound: all voctors
generalizedinverses,
thestandardcoordinatesystem.On the otherband,the book coverssuchrelatively esotericthingsas-to-norms,
is introducedthroug! an apandKroneckerproducrs.The approachis mindful of the beginner.Wheneverpossible,a new concept
is first definedfor two vectors,thedeterminant
plication, or a numericale*"mpii, or a geometricalinterpretation.Linear dependence
eafly
chapters.chapter 1 is a leisrnelyintfoduction
in
the
true
is
trii
aspecially
in
R2.
for 2xlmaEices, andlinear transformations
asgraphsandincidencematrices,andconsuch
examples
by
motivated
and
aray
as
an
to matrices,a matrix being definedmerely
introducedslowly andconcretely.
operations
mabix
with
vein,
in
this
starts
2
also
chapter
nectivity matricesin the s6cial sciences.
rnentioned
so early: Hadamudpro<tnot
usually
with
topics
here
apparent
is
alreacly
The specialconcernfor statisticalapplications
first of many
ucts, idempotentmatrices,or muttipiication of partitionedmatrices.The Kroneckerproductis discussedhere for the
jump
in the level
marls a sudden
times,andthe treatmentof it is surprisinglydetaited.chapter 3, dealingwith linear independence,
of its row+chelon form' The
of abstraction,eventhoughttrerant of a marix is definedratherconcretelyin teflns of the appearance
moreconsaete:it deals
treagnentdeepensin Chipter 4, which coversvector geomery. This chapteris at oncemore conceptualand
It is a denseand
with abstractvectorspaces,nofins andscalu producis,but thesenotionsaregivenappealinggeometricmeaning.
orthogonalsubspaces,
difficultchapter,though,with severalnotionscascadingwithin a shortspace:spanningset,basis,subspace,
formalmathematics'
avoids
resolutely
book
the
since
vague,
little
a
necessarily
are
the
definitions
Someof
orthogonalcomplements.
is denotedby
the
operation
and
union
is
a
it
called
is
treated:
spaces
vector
of
two
sum
the
is
way
A regtettableinaccuracyhere the
dealswittt
concerns,
sense.chapter5, reflectinga statisticians
u. Thereis no indicationthat this is not a unionin the set-theoretic
propertiesof traces,resultsare
what the authorcalls threematrix reductions:trace,determinant"andnorm. In additionto the usual
statedfor idempotentmatricesand for the traceof a Kroneckerproduct.
of uy order. A
The determinantis defined only for 2x2 mttices, and a result is given for diagonaland triangular maFices
=
oo. In Chapter
p
given
I
2,
and
fot
-norms
,
aredefined,andexamplesare
nonsingularmatrix is definedin termsof its determinant..Lo
of
apaflitioned
of
aKroneckoproduct,
ofinversesuesiated,includingrulesfortheinverse
6, onmatrixinversion,themainproperties
will be both
discussion
that
the
states
marix, and of specialmatrices.Generalizedinversesareintroduced,andalthoughthe author
properties;
one;
several
for
computing
brief andelementary,thereis plenty there: ageneralform of a generalizedinverse;analgorithm
linear
transformation'
7,
on
Chapter
anda discussionof specialtinds or inverses(reflexiveinversesandtheMoore-Penroseinverse.)
in R2. Useful notionssuchas
is a returnto simple,concretemathematics:small matrices,graphs,and geometricinterpretations
in Chapter
models,discussed
linear
for
andobliquerotationandorthogonalprojectionsareillusnated.Thispavestheway
orttrogonal
meaning
geomerical
their
9,
with
in
Chapter
g along with simultaneousequations. UgenvAuei ano eigenvectorsare introducecl
product
it
is
the
last
def,ned:
is
at
matrix
amptydemonstrat€din n2. tiris is where,ostensibly,the determinantof an afbitfary square
result,
presented
a
as
here
5,
is
promised
in
chapter
of its eigenvalues.The readermay feel cheated;especiallysincethe definition,
deals
also
This
chapter
avoided.
are
of this book that detaitstoo technical
not a definition. But it is in reepingwith the appr-oacn
10
Chapter
decomposition.
singular-value
the
the maximumanOtheminimumof quadraticforms;and
with matrixdiagonalization;
intuitively'
gradually
and
it
to
up
It
builds
ellipsoid.
aimsessentiallyat definingtheMahalanobisdistanceandthe minimumvolume
look at a varietyof topics.
so that the motivationfor the Malralanobisdistanceis nice ud clear.The final chaptertakesa cufsory
hodgepodgechapter'
this
to
is
relegated
matrices
Surprisingly,the important subjectof positive definite andpositive semidefinite
It coversmostof what a
All in all, the book is successfulat what it attemptsto do. It is elementarywithoutbeingmechanical.
untrained'
mathematically
to
the
accessible
notions
more
sophisticated
of
the
statisticalpfactitionerngedsto know, and makessome
algebrais
linear
of
knowledge
whose
undergraduates
statistics
namely,
of
readers,
I would alsorecommendthe book to anotherclass
or werelackingin intuitive
inadequatebecausethe courseor cowsesthey followed did not covertbetopicsmostusefulto statisticians
or geometricalcontent.
I Merris',MultilineorAlgebra
Russel
Montr6ol
Reviewedby S.W.Drury,McGillUniversity,
particularlyin thedomainsof Differential
MultilinearAlgebrais a subjectthathashadanenormousimpactoverthelastfew decades,
Marcus's monumentalFizite
GeometryandDifferential Topology. The textbook long reveredby the purists in the field is Marvin
have
Many developments
19751.
pp',
Dekker,
DimensionalMultilircar Atgebratprttt, x +292pp., Dekker,1973;Putll, xvi + 718
&
Gordon
Series,
Applications
and
takenplacesincethatboot ippeared andMuttitineir AlgebrabyRussellMerris [Algebra,Logic
a
textbook
use
for
as
suitable
Breach,vfii + 332pp.,ISBNg0-56gg-07g-0,lgg7l bringsthefield up to date.The book is especially
Merris coversPaflitions
in a graduatecouse in Multilinear Atgebra. The first foru chapterslay the groundwork. In the first two,
lmoge; No. 20
and Inner hoduct Spaces' He relatesboth of thesetopics to GraphTheory the former by developing
the degreesequenceof the
graphandthe latter by snrdyingthe specral theoryof the Laplacianmatrix of a graph.
mii is a featureof the booh whereverrhere
are connectionsto Graph Theory,Merris developsthem. This is refteshingueciusl the empbasiscoming
ftom multilinear atgebra
is slightly different from that usually drawn betweenGrapbTheory anopritrix Theory. Permutationgroups
andthe Representation
Theory of Finite Groupsform the subjectmatterof the third and fourttr chapters,aod o"y are treatedin
a more-or-lesstraditional
fashion. Then, at last, comesa very thoroughchaptercovering the basicsof TbnsorSpaceswithout being too
clry. The next two
chaptersdevelopthe theory of Symmery Chssesof TbnsorsandGeneralizedMatrix Functions.The treatmentof these
topicsis the
only availablediscussionin book form that is truly up to date.Nevertheless,nnny of the morerecentresultsareleft for
thereaderto
trackdown onhis own, the text notreally contributingsubstantiallyto thereader'senlightenrrent.The SymmetryClasses
chapterends
with a nice genoalization of P6lyaEnumerationtogetherwith a frrrtherapplicationto NMR Specroscopy.The GeneralizedMatrix
Functionschapterdescribesthe outstandingconjecturesof the subjecuthe Permanent-on-Top
Conjectureandthe soulesconjecture,
anddetailsrecentprogless.Finally thereis a chapteron the RationalRepresentations
of GL(n,c), togetherwith an applicationto
the graphenumerationproblem. This is a nice book, stimulatingto readandexcellentboth as a referencebook andasalextbook for
a graduatecoruse.
Fuzhen Zhong: LinearAlgebra:challengingproblemsfor students
Reviewedby WosinSo,SomHoustonStoteUniversity,
Huntsville
Lircar Algebra: Clnllenging Problemsfor StudentsbyFuztrenZhangpohns Hopkins Studiesin the MathematicalSciences.
Johns
HopkinsUniversityPress,xii + 174pp., ISBN 0-8018-5458-x(H), ISBN 0-E018--545g-g
(p), 19l)6lis a collecrionof 200miscellaneousproblemsin linearalgebrawith hintsandsolutions.Sincethebookis notintendedto be a text,but only a supplement,
thereader
is assumedto haveat least a fust coursein linear algebra.Someproblems,however,do demandmoreknowledgethan
a fust course
will provide,for instancethe Jordanform andeigenvalueinequalities.The bookis split into two parts:(l) problems,(2) Hints
an<l
Solutions.For easyreference,the 200problemsareroughlydividedinto five chapters:l. VectorSpaces,
2. Determinut, Inverses,
Rank,andLinearEquations,3. Matrices,LinearTransformations
andEigenvalues,
4. SpecialMatrices,urd 5. Innerhoduct Spaces.
The selectionof problemsis basedpurelyon theauthor'staste.PaulHalmosoncesaidthatsolvingproblemsis fun provided
thatthe
problemsareneithertoo hard nor too easy.Overall the reviewerbelievesthat the authordoesprovide a lot of fun for
the reader.To
achievea certainelegancy,however,the hints andsolutionstendto lack explanation.The reviewerhasfoundthebook
to be very
usefulin writing examsfor a linearalgebracourse.It is hardto write a perfectbook,but the authorhasmadea list of known
enors
andmisprintsavailableat the websitehttp: / /rr'ww.polaris . nova . edu/-zhang.
New & ForthcomingBookson LlneorAlgebro & RetotedTopics:I gg7-1ggg
Listedbelowaresomenew andforthcomingbookson linearalgebraandrelatedtopicsthathavebeenpublishedin 1gg7
or in lggg,
or arescheduled
for publicationin 1998;(P) denotespaperbackand(H) hardcover.This list updatesandaugmentsour Gurde
[69].
Thereferencesto CMP = CurrentMathematical PublicationsandMP.=MathematicatReviewiwe,refoundusingMathsciNet
andto
Zbl=ZearalblattfLrMathefiwtik/MATHDfuaba.rebyvisitinght'tp:
/lww,t.emis.de/csi-bin/t[AIlt(tlrreefreehitsareallowed
for non-subscribers).
The following public websiteswerealsohelpful:
Amazon-"Eafthsbiggestbookstore": http: //www.amazon.com
Barnes & Noble: http
: / / shop. barnesandnoble
. com/BookSearclr/
search
COPAC:OnlinePublicAccessCatalogue[to someof thelargestuniversityresearchlibrariesin theUK andlreland]:
h t t P : , / / c o P a c. a c . u k / c o p a c /
Interloc: "The worlds oldest and largestdatabaseof out-of-print, rare, and antiquarianbooks": hrrp :
/,/dani e1 . inrert
Libraryof CongressCatalogs:BrowseSearch:hrtp://lcweb.loc.govlcara1og/browse/
Melvyl System:The university of california: http: //www.mel'yl .ucop.edu/
oc . com/
TheNewYorkhrblicLibrary-CATNYRTheResearchLibrariesOn-LineCatalog:hrrp: //1as.123.101.18/search/.
Many thanks go to Simo hrntanen and G0tz Tfenkler for their help. All additions and corrections are most welcome
ancl should be
sent by e-mail to George Styan at styangMarh. McGill . cA.
poge
Aprll 1998
References
tll
Apostol, Tom M. (1997). Linear Algebra: A First Course
w ith App I ic at ioru to D ffi rert iol Equations. Wiley, ISB N 0 47 | 17421-1,in press.
tzl Axler, Sheldon (1997). Linear Algebra Done Right. 2nd Edition. Undergraduate Texts in Mathematics, Springer, xii + 251
pp., ISBN 0-387-98258-2(P), ISSN 0172-6056,Z;bl970.42297.
(OV: 1995, 23Epp.)
t3l Baltagi, Badi H. (199t). Econometrics.Springer,398 pp., ISBN
3-5N-63617-X (P). (Solwions Manual, 321 pp., ISBN 3-5n638e6-2(P).)
t4l Bandle, C.; Everitt, W. N.; l.osonczi, L. & Walter, W., eds.
(1997). Gencral Inequalities-7: Proceedings of the 7th Intemational Conference held in Oberwolfach, November 13-18,
I99S.International Series of Nurnerical Mathematics, vol. 123,
Birkbluser, xii + 4Bpp., ISBN 3-7&3-5722-3 (H), ISSN 037331 49, MR 97m:0fi)25, Zbl 864.00057.
t5l Belhnan, Richard (1997). Introduction to Matrix Analysis.
Reprint of the Second Edition. With a Foreword by Gene H.
Golub. Classics in Applied Mathematics, vol. 19, SIAM, xxviii
+ 403 pp., ISBN 0-89871-399-4 (P), CMP l:455-129, Zbl
872.15003.(OV: 1970, McGraw-Hill, MR 4l:3493.)
t6l Blackford, L. S.; Choi, J.; Cleary, A.; D'Azevedo, E.; Demmel,
J.; Dhillon, I.; Dongana, J.; Hanrnarling, S.; Henry, G.; Petitet,
A.; Stanley, K.; Walker, D. & Whaley, R. C. (1997). Sca[,APACK Users' Guide. SIAM, xxvi +325 pp.& I CD-ROM, ISBN
0-8987t-397 -8, Zbt 970.34142.
t7l Blyth, Thomas Scott & Robertson, Edmud Frederick (199E).
Basic Linear Algebra Springer Undergraduate Mathematics Series, Springer, ix + 2Ol pp., ISBN 3-5N-76122-5 (P), Zbl
980.06309.
t8l Bultheel, Adhemar & Van Barel, Marc (1997). Linear Algebra,
Ruional Approximation and Ortho gonal Polynomials. Studies
in Computational Mathematics, vol. 6. North-Holland, xviii +
U6 pp., ISBN 04U-82872-9, CMP l:602-017.
t9l Carniz, S. & Stefani,S., eds. (1996).Matrices and.Graphs: Tlw-
5
tt4l
Davis, Artrice (1998). Linear Circuit Anolysis.PWS, 1072 Pp.,
ISBN 0-534-95095-7, in press.
tl5l
Davis, M. H. A. (1997\. Linear Stocho*dcSystems.2nd Edition.
Chapman& Hall, 256 pp., ISBN 0412-31740-0 (H),in press.
tl6l
Demmel, JamesW. (1997). Applied Numerical Linear Algebra.
SIAM, xii + 419pp., ISBN 0-89871-389-7(P),2b1970.37451.
tlTl
Diestel, Reinhard (1998). Groph Tlwory. Translated from the
German. Graduate Texts in Mathematics, vol. 173, Springer,
(H), ISBN 0-387-98211-6(P),ISSN W72ISBN 0-3E7-9E210-8
5285, ZbL 873.05001, in press. (OV: 1996, Graphentheorie,
Springer,xiv + 2N pp., ISBN 3-540-60918-0,2b1859.05001.)
t18l Draper, Nonnan R. & Smith, Harry (1998). Applied RegressionArulysu. 3rd Edition. Wiley,ISBN O47l-17082-E, in press.
(OV: 1966,ix + 407pp., MR 35:Ml5; 2nd Edition: 1981,xiv +
7W pp., MR 82f:62OO2.)
tlgl
Farebrother,R. William; Puntanen,Simo; Styan,GeorgeP. H. &
Werner, Hans Joachim, eds. (1997). Sixth Special Issue on Linear Algebra ard Statistics. Linear Algebra and Its Applications,
vol. 264,506 pp., ISSN 0024-3795.
t20] Farin, Gerald & Hansford, Dianne (1997). Tlte Geometry Toolbox for Graphics and Modcling. A. K. Peters, 300 pp., ISBN
l -5688r-074-r (H).
I2ll
Flnry, Bernard (1997). A First Course in Multivariate Statistics.
SpringerTextsin Statistics,Springer,xiii +713 pp.,ISBN 0-3879E206-X (H), Zbl 970.46390.
I22l Friedberg, StephenH.; [nsel, Arnold J. & Spence,Lawrence E.
(1997). Linear Algebra 3rd Edition. Prentice Hall, xiv + 557
pp., ISBN 0-13-233859-9 (H), CMP l:434-M4; International
Edition, Prentice-Hall lnternational, ISBN 0-13-273327-7 (P),
ZbilE6/.l5001.(OV: 1979,xiv + 514 pp.,MR 80i:15001
,Zbl
44E15003;2ndEdition:1989,xiv + 530pp.,MR90i:I500l ,Zbl
69s.l5oor.)
I23l Godunov, S. K. (1997). Modcm Aspects of Linear Algebra frn
Russianl.NauchnayaKniga Novosibirsk, xxv + 390 pp., ISBN
5-8El 19-013-0,2b19ffi.42754. (Translatedas [24].)
I24l Godunov, S. K. (1998). Modcrn Aspects of Linear Algebra.
Translated from the Russian t23l.Tlanslations of Matbematical
Monographs,vol. 175. AMS, in press.
j:{:i"f
iYi;
27::f
,ii*'i#'",;,:::;;":fr'::;,i::;,i:"f
June 1995.World Scientific, 256 pp., ISBN 981-02-3038-9.
tlOl Carlson, David; Johnson, Charles R.; L^y, David C.; Porter,
A. Duane; Watkins, Ann & Watkins, Williatrt, eds. (1997). Resourcesfor Teaching Linear Algebra. MAA Notes, no. 42.
ldAA, x + 287 pp., ISBN 0-88385-1504 (P), Zbl 868.15004.
tlll
Censor, Yair &Zentos, Starnos A. (1997). Parallel Optimization: Thcory, Algorithms, and Applicaions. Nuurerical Matbematics and Scientific Computation, Oxford University Press,
576 pp.
Uzl Cullen, Charles G. (1997). Linear Algebra Wth Applicuioru.
2nd Edition. HarperCollins, 432 pp., ISBN 0-673-99386-8(H),
2b1970.42333.(OV: l9EE, 393 pp.)
tl3] Cvetkovi6, D.; Rowlinson, P. & Simii, S. (1997). Eigenspoces
of Graplu. Encyclopedia of Mathematics & Its Applications,
vol. 66. CarnbridgeUniversity Press,xiii +258 pp., ISBN 0-521-
57352-r.
l25l Greenbarm, Anne (1997). Iterative Methodsfor SolvingLinear
Systems.Frontiers in Applied Mathematics, vol. 17. SLAM, xiv
+ 220 pp., ISBN 0-E9871-396-X(P), Zbl 970.37450.
t26l Gmber, Marvin H. J. (199t). Improving Efficiency by Shrinkoge: Tlrc Jarnes-Steinatd Ridge RegressionEstimators.Statistics: Textbooks and Monographs, vol. 156, Marcel Delker, xii +
632pp., ISBN 0-8247-0156-9(H).
I27l Gunawardena,Ananda D. et aI.(1998). InteractiveLinear Algebra With Maple V: A Complete Software Packagefor Doing Linear Algebra. Tims: Textbooks in Matbematical Sciences(P), in
press.
t28l Hall, Frank; Pothen, Alex; Uhlig, Frank & Van Dooren, Paul,
eds. (1997). Proceedings of the Ftfih Confererce af the Intenntional Linear Algebra Society: Held at Georgia State University,
Atlanta, GA, August 1t19, 1995.Linear Algebra and lts Applications,vol. 254,vii + 551 pp.,ISSN 0024-3795,MR 97r:15003,
zbl 865.00040.
poge
6
lmage; No. 20
l29l Hanselman,Duane & Littlefield, Bruce (1997). Tlrc StudcntEdition o/ MATI-AB@, Version 5, (Jserb Guide. hentice Hall,
xxxix + 429 pp., ISBN O-t3-272550-9(p).
t43l l(hattree, Ravindra & Naik, Dayanand N. (c. March lggT). Applied Multivariate Srcrisrics Wth SAS Software. SAS Insriture,
400 pp., ISBN l-55544239-0 (H), in press.
t30l Hansen, Per Christian (199E). Ra*-deficient and Discrete IIIposed Problems: Numerical Aspectsof Linear Inverslon. SIAM
Monographs on Mathematical Modeling and Computation, vol.
4. SIAM, xvi + 247 pp., ISBN 0-89821-403-6(p), cMp l:4g6577.
l44l Kleinfeld, Erwin & Kleinfeld, Margaret (c. lggT). A Stwrt
course in Matrix Tlwory.Nova science, vi + l5g pp. & I IBMPC floppy disk (3.5 inch; HD), ISBN L-S6O7Z-4ZZ-6
(H), CMp
| : 442-036, Zbl 97O.37513.
[3 U Harville, David A. ( 1997\. Matrix Algebrafrom a Statistician's
Perspective. Springer, xviii + 630 pp., ISBN 0-3gz-9497t-x
(H), CMP l:467 -237, 2b1 970.46298.
I32l Har:uA. K. (1998). Matrix: Algebra, Colculw andGencralized
Inverse, Cambridge lnternational Science Publishing, 3 vols.,
l24O pp., ISBN l-898326-541, in press.
t33] Heath, Michael T.; Torczon, virginia; Astfalk, Greg; Bjoeorstad,
Petter E.; Karkp, Alan H.; Koebel, Charles H.; Kumar, Vipin;
Lucas, Robert F.; Watson, Layne T.; Womble, David 8., eds.
(1997). Proceedings of the 8th SIAM Corfererrce on Parallel
Processingfo r Scientifrc Computing : Minneapolis, March I gg7,
SIAM, CD-ROM, ISB N 0_ggg7 | _395_t, zbt 970 .37475 .
[34] Helmke, Uwe; Pritzel-Wolters, Dieter &Zen Eva, eds. (1997>.
Operators, Sysrents,and Linear Algebra-Three Decadcs of Atgebraic SysremsTlwory: Dedicated to PaulA. Fuhrmann onthe
Occasion of his 6Ah Birthday. EnropeanConsortium for Mathematicsin Industry.B.G. Teubner,zz4pp.,ISBN 3-519-02609-2,
zbt 873.00030.
t35l Hoffman, Kenneth & Kunze, Ray Alden (1998).Linear Algebra.
3rd Edition. PrenticeHall, in press.(oV: 1961,ix + 33zpp., MR
23:A3146;2ndEdition: 197t, viii + 4O7pp.,MR 43:199g.)
t36l Hsiung, C. Y. & Mao, G. V. (1998). LinearAlgebra.World Scientific, 300 pp., ISBN 981-02-3092-3 (H), in press (expecred
Sumner 1998).
t37l Huhtanen, Marko ( 1997). Numerical Linear Algebra Techniques
for Spectral Approxfunation and Applications. Dissertation, Research Report A380. Helsinki University of Technology, Institute of Matbematics,Espoo,ISBN gsl-zz-3s46-3, CMp l:4g0-
r7t.
t38l Johnson,I*e w.; Riess,R. Dean & Arnold, Jimmy T. (199g).
Introduction to Linear Algebra. 4thEdition, Addison-Wesley, xv
+ 551 pp., ISBN 0-201-824t6-7(H), zbt97o.4zs66.(ov: 1981,
ix + 358 pp.; 2nd Edition: 19E9,xi + 57Epp.; 3rd Edirion: 1993,
)k 576 pp. StudentsSolution Manual,ISBN 0-201-M276-0 (p),
in press.)
t39l Jprgensen, Bent (1997). TTrc Tlwory of Dispersion Modcts.
chapman & Hall, xii + 237 pp., ISBN 0-4l 2-9971l-g, cMp
l:462-891.
t40l Kaki[613, Yuichiro (1997). Muttidimercional Second Order
Stochastic Processes. Series on Multivariate Analysis, vol.
2, world Scientific, x + 3u pp., ISBN 9gl-0230o0-l , zbl
970.42489.
[41] Kapur, J. N. (1997\. Inequalities: Theory, Appticaions and
Measurements. Mathematical Sciences Trust Society, New
Delhi, vi + 242pp., CMP 1:480-914.
I42J Kaye, Richard & Wilson, Robert (199E). LinearAlgebra. Oxford
University Press,IsBN 0-19-850238-9(H), 0-19-850237-o(p),
in press.
[45] Knowles, JamesK. (1998). Linear Vector Spacesan^dCartesian
Teruors. oxford University Press, viii + 120 pp., ISBN 0-195tt2s4-7 (H), Zbt 980.0I 705.
[46] Kcicher, Max (1997). Lineare Algebra urd analytische Geometrie [in Germon].Fourtb Expanded and Revised Edition.
springer-Lehrbuch. springer, xiii + z9l pp., ISBN 3-s40-629033 (P),Zbt877.15002.
I47l Kwak, Jin Ho & Hong, sungpyo (1997). Linear Algebra.
Birkh[user, ix +369 pp., ISBN 0-8176-3999-3(H), cMp t:456992, Zbt 970.42497.
t48l Lehoucq, R. B.; Sorensen,D. c. & Yang, c. (1998). ARPACK
Users' Guide: Solwion of Large-Scale Eigenvaluc Problems
with Implicitly Restarted Arnoldi Methods SIAM, approx. 140
pp., ISBN 0-E9E7l-407-9 (P), in press(expecredApril 1998).
t49l l-eon, Steven J. (1998). Linear Algebra with Appticatiotu. 5th
Edition. PrenticeHall, xiv + 491 pp., ISBN 0-13-849308-l. (ov:
1980,xii + 338 pp.; 2nd Edition: 1986,xv + 408 pp.; 3rd Edition:
1990,xv + 458 pp.; 4th Edition: 1994,Macmillan, xv + 508 pp.)
l50l Li, Liangqing (1997). Classification of Simple C* -Algebras: Indrctive Limits of Matrix Algebras over Trees. Memoirs of the
American Mathematical Society,vol. 605, AMS, lB pp.,ISBN
0-82I 8-0596-7 (p), IS SN 0065-9 266, Zbt 97O.3 64t7 .
t5II Lidl, Rudolf &Prlz,Gunter (199E).AppliedLinearAlgebra. Znd
Edition. Springer,486 pp., ISBN 0-387-98290-6.
t52l Lind, De0ef (1997). Koordinaren, Veldoren, Matrizen: Einfiihrung in die lineare Algebra und anolytische Geometrie lin
GennanJ. Mathematische Texte, Spektrum Akademischer Verlag,ix + 271pp.,ISBN 3-8274-0193-3
(P),cMp l:453-885,zbl
868.15001.
t53l lnwdin, Per-olov (1998). Linear Algebra for euantum Theory.
Wiley, ISBN A47l-19958-3 (H), in press.
[54] Mathai, A. M. (1997). Jacobians of MatrixTransformations and
Functions of Matrix Arguntent. World Scientific, xii + 435 pp.,
rsBN981-02-309s-8
(H).
t55l Mathar, Rudolf (1997). It[ultidimercionale Stcalierunt [in Germanl. MathematischeGrundlagen und algorithmische Aspelre.
Teubner Shipten zur Mathematischen Stochastik. B. G. Teubner, Stuttgart,95 pp., ISBN 3-519-02738-0,cMp l:473-223.
t56l Maz'yu vladimir & Shaposhnikova,Tatyana (1998). Jacqucs
Hodamard: A Universal Mathematician. History of Mathematics, AMS, 574 pp., ISBN 0-8218-0841-9,
in press.
t57l Merris, Russell (1997). Multilirwar Algebra. Algebr4 Lngic &
Applications, vol. 8. Gordon & Breach,viii + 332 pp., ISBN 905699-078-0(H), IssN I 041 -5394,Zbt 970.374zs. (Reviewedin
thts Image 2O:34.7
[58] Milovanovi6, G. V., ed. (1998). Recen Progressin Inequalities:
Dedicated to Prof. Dragoslav S. Mitrircvi|. Mathematics and
its Applications, vol. 43o. Kluwer, xii + 519 pp., ISBN a-79234845-l (H), Zbt E82.0oot3.
April 1998
poge 7
l59l Mciller, Herbert (1997\. Algorithmische lineare Algebra: Eine
Einfilhrung frr Mathematilccr und Informatiker [in GemranJ.
Vieweg, Wiesbaden, x + 3E9 pp., ISBN 3-52E-0552E-6,Zbl
865.15001.
t73l Redheffer, Ray M. (1997\. Linear Algebro. Mathematics Series.
Jones & Bartlett, ISBN 0-8672-0288-2 (H), in press.
[60] Moore, Hal G. & Yaqub, Adil (199E). A First Course in Linear Algebra 3rd Edition. Academic Press,ISBN O-12-5057ffi-l
(H), in press. (2nd Edition: 1992, HarperCollins, xv + 551 pp.)
t75l Searle,Shayle R. (1997\. Iinear Modcls.Wiley ClassicsLibrary
[Reprint Edition]. Wiley, xiv + 532 pp., ISBN 0471-1E499-3
(P), MR 98e:62102.(OV 1971,MR 45:2E6E.SolwionsManual
available from the author: Biometrics Unit" Comell University.)
[6U Mukherj€€, S. P.; Basu, S. K. & Sinha, B. K., eds. (1998). Frontiers in Probability and Statistics. Narosa Publishing House,
New Delhi, x + 379 pp., ISBN 8l-7319-l l5-8. (38 selectedpapers presented at the Second International Triennial Calcutta
Symposium on hobability and Statistics, December 30, l99L
January 2, 1995.)
162l Nakos, George & Joyner, David (1998). Linear Algebra Wth
Applicaiow. PWS, ISBN 0-534-95526-6(H), in press.
t74l Schay, Gezt (1997). Linear Algebra. Jones & Bartleq ix + 303
pp., ISBN A-86720-498-2(H), Zbl872.15002.
176] Singh, Surjeet (1997). Linear Algebra. Vikas Publishing House
Pvt, Ltd., New Delhi, viii + Uzpp.,ISBN 81-259-04824, CMP
l:488-962.
[77] smith, I-arry (1998). Unear Algebra 3rd Edition. Springer,
ISBN 0-3E7-98455-0,in press. (OV: 1978, vii + 280 pp., MR
57:9712; Znd Edition: 1984, vii + 362 pp., MR E5h:15006.)
[63] Okninski, Jan (1998). Semigroupsof Matrices. Series in Algebra, vol. 6, World Scientific,32Opp., ISBN 9El-02-3445-7,in
press (expectedSumner l99E).
t78l Springer,Tony [Tonny Albertl (199E). Lineor Algebra.Znd Edition. Progressin Mathematics, vol. 9. Birkhluser,ISBN 0-81764O2l-5 (H), in press.(OV: l98l: Edited by J. Coates& S. Helgason,x + 304 pp., MR 84i:20002.)
t64l Parlett, Beresford N. (1998). Tlw Symtetric Eigenvalue Problern corrected Reprint Edition. SIAM, xxiv + 398 pp., ISBN 089871-N2-8, cMP l:490-034. (ov 1980, Prentice-Hall,xix +
348 pp., ISBN 0-13-880047-2,MR 8lj:65063.)
I79l Steeb, Willi-Hars, in collaboration with Tan Kiat Shi (1998).
Matrix Calculus ard Kroruclccr Product with Applicatioru to
C++ Prograrns. World Scienti0c, ISBN 981-02-3241-1, in
press.
t65l Penney,Richard C. (199E). Linear Algebra: Ideas and Applications. Wiley, xvi + 382 pp., ISBN 0471-18179-X.
[80] Stewart, G. W. (1998). Matrix Algorithms: Basic Decompositions SIAM, approx.4ffi pp., ISBN 0-89871-414-l (P), in press
(expectedAugust 1998).
t66] Plesken,Wilhelm (1998). Linear Pro-P-Groups of Finite Width.
Springer,ISBN 3-540-63643-9(P), in press.
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Lineare Algebra
urtd Anwendunten [in GenrranJ. Fachbucbverlag l*ipzig im
Hanser Verlag, Milnchen, 3?E pp., ISBN 3446-IE7OZ-2, Zbl
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t6El Puntanen, Simo (199E). Matriiseja filastotieteilijtille [in
Finnishl. Preliminary Edition. Report B4l , Dept. of Mathematical Sciences,University of Tirmpere, viii + 443 pp., ISBN
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to Boolcs on Matrices ard Books on Inequalities, with Statistical and Other Applicaiorc: Preliminory Edition In A Bibliograplry and a Guide, prepared for the Sixth lnternational
Workshop on Matrix Methods for Statistics flstanbul, August
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t7ll
Rao,C. Radhakrishna&Rao, M. B.(1998). MatrixAlgebra an"d
Its Applicuions to Statistics and Econometrics. World Scientific, approx. 600 pp., ISBN 981-02-3268-3,in press (expected
Summer 1998).
I72l Rao, Poduri S. R. S. (1997). Variance ComponentsEstimation:
Mixed Models, Methodologies and Applications. Monographs
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[8 U Straurpp,Walter (1997 ). Hilhcre Mathematik rnit MathematicaBand I: Grundlagen, Lineare Algebra [in GennanJ. Lehrbuch
Computeralgebra.Vieweg, Wiesbaden, viii + 313 pp., ISBN 3528-06788-8(P), Zbl 865.00m7.
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in press.
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[84] Trefethen, Lloyd N.& Bau, David m (D97). Numerical Linear Algebra. SIAM, xii + 361 pp., ISBN 0-89871-361-7 (P\,Zbl
874.65013.
lEsl Van Huffel, Sabine, ed, ( 1997). RecentAdvances in Total l*ast
Squares Techniques an"d Enors -in-Variables Modcling : Proceedings of the 2rd International Wo*shop on Total l*ast
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(P),CMP l:441-457.
t86] Werner, Wilhehn (1998). Mathematik lernen mit Maple: Ein
Lchr- und Arbeitsbuch fiir das Grundstudium-Integralrechntutg, Dffiretialgleichungen,
Lineare Algebra [in GennanJ.
2nd Revised and Enlarged Edition, with CD-ROM. Heidelberg,
633 pp., ISBN 3-920993-94-2(P), Zbl 980.05833.
l87l Xu, Shu-Fang (1997). Algebraic Inverse Eigenvaluc Problems
[in GennanJ.Vieweg, Wiesbaden,330 pp., ISBN 3-528-066849,Zbl970.09244.
user-friendlq and modern
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each additional book.
Outside USA/Canada:Add $4.50per book.
All overseas dellvery ls via airmall.
poge I I
lmoge,'No.20
28,1998
Report:MorchI ,1gg7-Februory
fLAS
Treosurer's
of WestFlorido, Pensocoro
by JomesR,Weover,University
Ealance on bans! Marc! L- _1_9-9_?
C e r t i f r c a t e o f D e p o s i t s 1 C O)
Vanguard
qheeking
Marqh
Account
BalarLce on March L
19,oo0.o0
8 , 34 2 . 8 4
2 2 , 3 r 0. 4 4
1J97
L997
I ncome :
Dues
780.00
I n t e r e s t ( G e n)
155.94
Contr ibut ions
General Fund
52.00
Hans Schneider prize
42.00
OT/JT Lecture Fund
2,00
F. Uhlig Pund
12.00
Conference Fund
2.00
ILAS/LAA Fund
1000.00
Expenses:
Sec. of State
70.00
Aor i I 1997
I ncome:
Dues
2 2 0 .0 0
I n t e r e s t o n C D ( H SI
74.56
Interest on CD (Of / Jt y
2 9. 0 0
I n t e r e s t o n C D ( F UI
23.LL
I n t e r e s t ( G e n)
14.99
Contr ibut ions
General Fund
50.00
OT/JT Lecture Fund
20.00
Expenses:
Postmaster (Stamps & Mailingt
40.00
Lisa M. Weaver (Update files ) 49. 59
Mav L997
f neome:
I n t e r e s t ( G e n)
1 4. 6 6
Expenses:
J u d y K ' W e a v e r ( A n n. T r e a . R p t ) 1 0 8. 5 0
June rg97
I rrcomet
Dues
290.00
I n t e r e s t ( G e n)
r 5 2 . 4g
E x p e n c e s:
00..00
Julv 1997
I ncome:
Dues
320.00
I n t e r e s t ( G e n)
t4 .64
I n t e r e s t o n C D ( H S)
75.39
Interest on CD (Ot/ JT )
2 9. 3 2
I n t e r e s t o n C D ( F UI
23.63
E x p e n s e s:
Producing & Mailing "IMAGE"
George P.H.Styan
1329.73
Office Depot ( Supplies )
4 8 .6 3
P o s t m a s t e r ( S t a m p s & B a l 1 o t ) 2 4 0 .t 5
Postmaster (Stamps-Dues
Notice I
Mail Boxes Etc. (400 Ballots )
Judy K .Weaver ( Ear ly Dues
Notice )
2.045.84
70.00
1,975.94
431.55
9 0 .1 0
3 4 1 .5 5
t4 .56
1 0 8 .5 !
( 9 3 . 8 9)
4 3 2. 4 9
oo. o!
432.49
4 6 2. 9 7
Z4b.18
25.69
108.50
1 ,996.gg
( 1 , _ 5 3 3 . 9)1
poge I 2
Auqust 1997
I n c o m e:
1 1 3t . 2 0
Dues
83.44
Interest (Gen)
Contr ibut ions
71.00
General Fund
85.00
Hans Schneider Prize
22L.00
OT/JT Lecture Fund
71.00
F. Uhlig Fund
L,7L4.64
51.00
Conference Fund
E x p e n s e s:
84.00
Judy K. Weaver (gallots )
Debit to accounting exchange
1 0. 0 6
6.05
r ate
September L997
I ncome:
1160.00
Dues
86.13
( G e n)
Interest
23.63
(
F
U
C
D
o
n
)
Interest
Contr ibut ions
20.00
General Fund
3
50.00
P
r
i
z
e
S
c
h
n
e
i
d
e
r
Hans
3 7 0. 0 0
OT/JT Lecture Fund
2 , O t g. 76
10.00
F. Uhlig Fund
ExPenses:
209.00
O f f i c e Depot ( SuPPIies )
&
(
R
e
c
o
r
d
i
n
g
JudY K W e a v e r
2 8 6. 0 0
u.00
UPdating Files )
October L997
I ncome:
15.80
15.80
( G e n)
Interest
Expenses:
35.31
t'lail Boxes Etc . (Ballots )
77L.24
P o s t m a s t e r ( S t a m P s)
&
JudY K. Weaver (Ballots
1 ,023.55
2r7 .00
t'tai l ings )
November L997
I ncome:
7 0 0. 4 4
Dues
84.24
( G e n)
Interest
76.23
(
H
S
)
C
D
Interest on
29.64
Interest on CD (OT/;t1
Cont r ibut ions
1.00
General Fund
25.00
Hans Schneider Prize
10.00
OT/JT Lecture Fund
931.55
5.00
F
u
n
d
i
g
U
h
l
.
F
ExPenses:
P o s t m a s t e r ( 2 n d D u e s - S t a m P s) 1 3 0 . ' l 6
77.00
JudY K. Weaver (l'!aifing)
2 0 8 .5 1
.75
C
h
a
r
g
e
Commercial Service
lmoge; No.20
L , 6 2 4. 5 8
1 ,733.76
( 1 , 0 0 7. 75 )
723.04
I
poge I 3
April I 998
December L997
I ncome:
7 6 0. 0 0
Dues
85. 58
I n t e r e s t ( G e n)
Contr ibut ions
3 0. 0 0
General Fund
1 1 0. 0 0
Hans Schneider Prize
5.00
Conference Fund
E x p e n s es :
Producing & Mailing "IMAGE"
1307.00
George P.H. Styan
33.25
Judy K. Weaver (IIAS Files)
3. 57
Service Charge
J_anuarv 19 98
I n c o m e:
840.00
Dues
15.54
I n t e r e s t ( G e n)
75.23
I n t e r e s t o n C D ( H S)
29.64
I n t e r e s t o n C D ( O T / J T1
I n t e r e s t o n C D ( F U)
23.37
,
Contr ibut ions
General Fund
40.00
llans Schneider Prize
1 1 0. 0 0
O T/ J T L e c t u r e F u n d
310.00
F. Uhlig Fund
1 1 5. 0 0
Conference Fund
10.00
Expenses:
Returned I tem Charge
20.00
S e r v i c e C h a r g e ( D e c)
4.32
F e b r u a r y 1 99 8 _
Income:
I n t e r e s t ( G e n)
15.09
Expenses
Service Charge ( Jan )
L0.79
990.58
1,34.3.92
L , 5 6 9. 7 I
24.32
GeneraL Fund
Frank Uhlig Educational
Hans Schneider Prize
Olga Taussky Todd/John
Conference Fund
1 .545.46
1 5. 0 g
10.79
**********************************************************************
Februarv 28, t998
Account Balance
Checking Account
Certificate
of Deposit (ru1
Certificate
of Deposit (Gen. )
Certificate
of Deposit
(72t Hs & 28t OTIJT)
Vanguard
(72t r{s & 28t orl JT )
( 3 5 3 . 2 4)
4 .29
27.702.7I
27,702.7L
1,500.00
10,000.00
7 ,500. 00
g,gg2.t5
55,594.96
22,L79.61
2,699.00
L4,735.35
Todd Fund
7,775.01
7 , L g 6. g g
ILAS/LAA Fund
1,000.00
55,584.96
***********************************************************************
Fund
qf,n fr',\l*
o
3/ashs
poge 14
lmoge; No.20
ThemoticTermon LineorAlgebroond ControlTheory
CoimbroAstronomicof
Observotory,
Moy I J-August7,1998
TheThematicTbrmon Linear Algebra an control rheory will beheld at
the coimbra Astronomicalobservatoryin thespring
of 1998:May ll-August 7, 1998.someresearcbers
will attendthiseventtbroughoutthefull threemonths,whilst otherswiu
participatefor periodsof one month or less. A weekly researchseminar
will be organized,led by eioer oneof thepermanent
membersor by guestspeakers.Therewill be two summerschoolsaimedat graduatestudents,
eachconsistingof two ten-hour
courses:
l. "StructureandDesign of Linear SysterN" by JeanJ. Loiseauandper Zagalak(May 2l-29).
2.'Lineu Algebra and Control Theory" by PaulFuhrmannandPeterLancaster(June lrZS).
To register,please,sendan e-mail with: Your narne,institution,e-mail and tbe indicationthat you wish to participate
in this
schooltolingEohermite.cii.fc.ul.pt.
Aregisradonfeeof15fi)0PortugueseEscudosisdueandmustbepaidatthe
school.A Worlshop will be held June 24-26,1998, whenmost short-stayresearchers
will be present.A limited numberof
15minutecontributedtalla will be accepted.Please,submityourtitle andabstractto 1in9 Bohlrnrire. cii . f c. u1 . pr .
Accommodation(SchoolsandWorlahop): Therearespecialpricesfor participantsat severalhotels.For informationabout
accommodation/reservations,
pleasecontact:Iurs. Liliana seabra cenuo lnternacionalde Matemdtica,complexo do observat6rioAstron6mico,
AlmasdeFreire,P-3040Coimbra,
Portugal;
FAX:351-39445380;
tin98ohermire. cii. fc. ul.pc.
Participantsconfirmedas of February9Eare(in alphabeticalorder): Itziar Baragaia(Univ. pais Vasco,SpainlMay; IsabelCabral(Univ. NovadeLisboa,PornrgalFMay/July;
JoanJ. Climent(Univ. Alicante,SpainFJunet2ndnAt;lose
A. Dias
da Silva (Univ. Lisboa / CIM, PortugalFMaylluly; Paul van Docen (CatholicUniv. Louvain. Belgiumpune/last week;
PaulFuhrmann(BenGurionUniv.,Israel)-May/July;JuanM. Gracia(Univ. PasVasco,Spainpune/lst half; DiederichHinrichsen(Univ. Bremen,Germany)-June/last
week;CharlesR. Johnson(CollegeWilliam an<lMary, U.S.A.punel2nd half;
Vladimir Kuera(Acad. Sciences,CzechRep.lMay Ll-17; PeterLancaster(Univ. Calagary,CanadatJune;JeanJ. Loiseau
(CNRS, Nantes,FrancelMay; Michel Malabre (CNRS Nantes,Frurce)-June/lastweek; Alexander Markus (Ben Gurion
Univ., Israel)-June2ndhalf; NancyNichols(Univ. Reading,UKpune/last week;Fernandohterta (Univ. Polit. Barcelona,
SpaintJune/July;J. F. Queir6(Univ. Coimbra"PortugallMay/July;JoachimRosenthal(Univ. Nore Dane, USAFJune;E.
Marquesde S (Univ. Coimbra, PortugatlMay-July; CarstenScherer(Tech.Univ. Delft, The NetberlandstJune/July;Hans
(CWI Amsterdam,TheNetherlands)-June,
Schumacher
lastweek;FernandoC. Silva(Univ. Lisboa PornrgallMay/July;Ion
Taballa(Univ. PasVasco,SpainfMay/July; Per 7-agalak(Acad.Sciences,CzechRep.lMay/June.
OrganizingCommittee:I. Zaballa(Univ. PaisVasco,Spain),Coordinator:
mepzatej ols. ehu. es, JoseA. Diasda Silva
(Univ.Lisboa/CIM, Portugal):perdiqaooher:nrite. cii . f c. uLpt, Fernando
C. Silva(Univ. Lisboa Portugal):f csilva
pleasevisithtrp: //herrnite. cii. fc.uI .pr/lin98/.
Gfc.u1 .pt. For furtherinformation
MotrixTheoryWorkshop
in Honourof ProfesorYik-Hoi
Au-Yeung
.|998
TheUniversity
of HongKong,Moy 15,
A Matrix TheoryWorkshop,in honourof theretirementof ProfessorYik-Hoi Au-Yeung,will be held in RoomG5, James
HsioungLee ScienceBuilding, Universityof HongKong, on FridayMay 15, 1998.The programmeof lhe workshopcomprisesof a numberof talks on variousaspectsof matrix theory.Tbntativespeakersinclude: Che-ManCHENG (Universityof
Macau),Siu-PorLAM (ChineseUniversityof HongKong),Chi-KwongLI (Collegeof lVilliam andMary,USA), Yiu-Tlrng
POON(IowaStateUniversity,USA), Lok-ShunSIU (Universityof HongKong),Bit-ShunTAM (ThmKurg University,Thiwan),Tin-YauTAM (AuburnUniversity,USA), Nam-KiuTSING (Universityof HongKong).
ContributedtalK arewelcome,subjectto availabilityof time slots.For fuftherinformation,pleasecontact:Dr. Nam-Kiu
TSING,Departmentof Mathematics,
TheUniversityof HongKong,PokfulamRoad,HongKong,Tbl: (852)2859-2251,
Fax:
(852)2559-2225,nkts ing@hku. hk.
April 1998
poge l5
SeventhConferenceof the Infernotionol
LineorAlgebroSociety
The"HonsSchneiderLineorAlgebro"conference
Modison,Wisconsin,
USA:June3-6, l99g
The 7th Conferenceof the IntemationalUnear Algebrasociety: The 'Tlans Schneider
Linear Algebra,,conferencewill provide an opportunity to bring togetherresearchersaml educatorsin all aspects pure
of
and eppli€dlinear algebraand matrix
theo'ryin order to allow for a broadexcbangeof ideasand dissemimtionof
recentdevelopmentsand results. The conferencewill be dedicatedto Hans schneiderin recognitionof his enormous
contributionsto line"r algebraand the linear algebracomrnrmity. The program as of Apil Zt,lgSg,rypears
on pp. lG2l below; the final prograol may be viewed as
http :,//rnath. wisc . edu/ trualdi /program. ps .
THEMES:Algebraic' aralytic andombinatorial nratrix theory nunrerical
lineanalgebra matrix pertlrbations,matrix stability andryplications in engineering'linear algebrain contrrol
*J ryrtoo" theory,splinesandliner algebrra,applicationsof
linear algebrato statistics,linear algibraeducaiion.
ORGANANG COMMIITEE: R. A. Brualdi (chair), B.
Cain, B. Datr4 J. Das da Silva S. Friodland,M. Goldberg,U.
Rothblum,J.Stuart,D. Szyld,andR. Varga
CoNFER'ENCEPRoCEEDINGS:A special.issueof the
iovnal linear Ngebyqand ltsApplicuions(LAA), with special
editorsB ' cain' B ' N' Datta tt't' cotdb""c' u. Rothblum,
andrj swro. ril" issu€will bedodiot"a to rt-, schneider.speakers
who wish to havetheir paperousidenedfor publication
in this .fod irr* of I-AA should submit rheir paperto one of the
specialedit'ors'only thosepapersthatmeetthe standeds
of L{A- andthl areapprovedby thenormalref.reeing proess will
be publishedin the spocialissue.Deadrinefor submission
is septenruerr0, 1998.
HousING: An entire (air-conditioned) dormitory,
chadbourne Hall, a two-minute walk to the mathematics
building Van Vleck Hall, has been reserved frr conference
puticipants. A special floor will accomndate couples.
Breakfast
is included in the room rate of $40 single and
$Zl per person in double room. Conference participants are
encotlraged
to stay at Chadbourne Hall. Rooms have also bem
set aside
at anearby Howard Johnson's Hotel. Those wishiog
to stay
at the Howard Johnson's should make their reservations
directly by calling: 608-251-5511. Mention the
conference
and the university to get the confere,ncerate of
$lq single
and $S4 double.
BAI.IQUET: A banquet will be held on Friday,
June 5,
1998, starting at 7 p^, in the Great Hall of the
Memorial
union (the union on the Terrace overlooking
Lake
Mendota) of {.IW-Madison. After dinner, several people
will
offer reminiscences and vignettes about Hans Schneidetr.
EXCURSION: An excursion is scheduled for Sahrrday
afternoon' June 6, 1998, to the famous'.House on the
Rock' in
nearby Spring Green. This is more than a fascinatin,g
house;
it is a veritable large rilrseum of remarlcablemechpnical
and
electrical contraptions with an incredible indoor
sarousel.
Not to be missed!
Von VleckHolr(photo courtesy:universityof wiscrcnsin)
woRKsHoP:
A Numerical Range workrlop will precele
the ILAS conferenceon June lA, 199g. For information
pleasecontact Chi-Kwong Li at ckliqnath.rur.edu
or visit
http z/ /wwtr-math. r^rr.edu/-ckli/wo'ra
. html .
poge 16
hnoge: No.20
Wednesday,
June3, 199E
9:0O-9:45am
9:5O-10:35
am
I l:05-l 1:30am
1l:35-12:00
noon
l2:0O-12:15
pm
B' Ross BARMISH: Risk-AdjusredLinear Algetua:A
New hracrigm
RAJENDRA BHAIIA: The positiveDefnitewss of Sorrrc
Marrices
RAPIIAEL LOEWY Chss FunctionhwEtalities
for PositiveSemidcfniteMatrices
PETERLANCASTER: TheInertia of Dissipaive
Quadruic Polynomiatsor ostowski-Schncider Revisited
ConferencePhon
2:0f2:25 pm
2:3V2:55pm
3:0G3:25pm
3:30-3:55pm
MINISYIIPOSIUM on Graphrheory andLinear Algeha I: orgmizedby
R. Maris
R. B. Bapa[ Unear estifitationin nwdelsbasedon a graph
PavelYu. Chebotarev:A graph interpretationof the Moore-Penroseimterseof
tltc l^aplacianmatrix
Onn Chan: Wrtex oricntuions and imnatwntal ircquatities
DouglasKlein: Intrinsic mctricson graphs,distarce fltatricesand graph iwariants
2:0f2:25 pm
2:3V2:55pm
3:00-3:25pm
3:30-3:55pm
MINISYMPOSIUM on TopologicalMethodsin Linear Algebr aI: OrganizedDyM. Goldbog
ShmuelFriedland: Thewncrcssing rulc revisited
MosheGoldberg: Not all polynomiatmappingsatc cottinuous
Roy Meshulam:Enrenml problemsfor spacesof matrices
Joel Robbin: Maslov indexand natrix tltcory
2:0A-2:15pm
2:2O-2:35
pm
2:4V2:55pm
3:00-3:20pm
3:25-3:45pm
MINISYMPOSIUM on E<lucationalIssuesin Linear Algebra l: Organized DyD. Culson andF. Uhlig
Luz Maria DeAlba: Teachingltnear hdependcnce,spu\ and bosis
David Carlson:Teachingspanand lfuear independencefirst
JaneM. Dty: Sonwwaysto providc variety and iwigltt
Richardo. Hill: stiu trying to kzeptlu fog at bay-not comptetelysucceeding
Thomas!V. Polaski: Howfoggy is it? Writing assignrrlents
and assessneilin tincar algebra
2:0U2:15pm
2:2V2:35pm
2:4V2:55pm
3:00-3:l5 pm
3:2V3:35pm
3:4f3:55 pm
CORELINEAR ALGEBRA I
R. Cant6: On the Segreclnracteristh of a block triangrlar matrix
WasinSo: Lircar opeiatorspresening thc numberof rcntrivial invariant polytomials
Enide Martins: Eigenvaluesof matrix communtursand Jordanprodrcts
J. D. Botha:Productsof diagorulizfuIematrices
Pei YuanWv Dilation to inflationsof S(6)
ReinhardNabben: Ttidiagonalnafiices and their inverses
3:2f3:35 pm
3:4V3:55pm
NI.'MERICAL AND COMPLN1ATIONALLINEAR ALGEBRA I
Daniel B. Szyld: Themysteryof asynchrotwusiterarionsconvetgencewhcn tlu spectralradius is one
Ivo Marek An aggregation/disaggregation
iterotion schemeusinggeneraloderings
Ljiljana Cvetkovid: Sone convergenceresuhsaboutahernuing iterarions
Karen S. Braman:Aspectsof the multi-shift QR-Algorithm:
Early deflationand maintainingwelhfocusedshifisto achievelevel 3 performuce
Heike FaBbendq:Sotru connectionsbetwcentlv SR/SZand HNHZ eigenvalw algorithms
DonaldW. Robinson:A usefulquality indicaorfor afinite elemcntcahulatbn
4:05-4:50pm
CARL DE BOOR:LirwarAlgebra IssuesinApproxittwtionTheory
5:00{:30 pm
Reception:9th Floor Lounge,VanWeckHall
2:0f2:15 pm
2:2W2:35pm
2:4V2:55pm
pm
3:00-3:15
Thursday, June 411998
8:30-9:l5 am GUERSHONHAREL: A DevelopnentalModel oI Students'Undcrstatdingof Proofs:
Historical, Philasophal, aad CognitiveConsiderations
9:2f10:05 am THOMAS J. LAITFEY SoneNew Inequalitiesinthe Nomegaive InverseEigenvahrcProblem
April 1998
poge 17
Thursday,June 4,1998 (conttd)
MINISYMPOSIUM on Graphrheory andLinear Algebra II: organizedby R. Merris
l0:35-l l:00 am Miroslav Fiedtet Additive compowtdmatricesand graphs
1l:05-ll:30 am SteveKirkland: Eftremizing algebraiccotnectivity subjectto graph ttvoretic constraints
I l:35-12:00am Alex Pothen:Distancemalrices,tlwir eigenvalrcsand vertexnumberingof graphs
10:35-10:50
am
10:55-1
1:10am
1 1 : 1 5 -l l: 3 5a m
l1:40-ll:55 am
12:0G12:20pm
12:25-12:4Apm
10:35-ll:00 am
I l:05-l l:30 am
I l:35-12:00noon
12:05-12:30
pm
10:35-10:50
am
l0:55-ll:10am
I l : 1 5 - l l : 3 0a m
I l:35-l l:50 am
1l:55-12:10
pm
12:15-12:30
pm
l0:35-10:50
am
10:55-1
l: l0 am
1 1 : 1 5 - 1 1 :a3m
0
1l:35-ll:50am
l 1 : 5 5 - 1 2 : 1p0m
l2:15-l2:30pm
2:0f2:45 pm
MINISY-MPOSIUMon EducationalIssuesin LinearAlgebraII: OrganizedbyD.Culson andF. IJhlig
R. Cant6:OncomputerMs inlinear algebra
cbristopher Beattte:fui on-line coursein tfuear algebrafor
first-year studens
Jeny Uhl: Mdrices, Geometryand Muhemuica
stevenJ. Leon: using sofrwareto visualizeeigenvaluesand singularvalues
A. Berman:Iirc ar trouformation s
Biswa N. Datta,:.Teachingof numericalliwar algebra
MINISYMPOSIUMon structuredMatrix problemsl: organircdby LotharReichel
DanielaCalveni: on an-iwerse eigenvatwprobtemfor Jacobi marices md applicaion
to
Gauss-Kronrod quadralttre ruIes
Biswa N. Dattz: Recentresultson panial pole and eigenstntctureassigtn entslor
matrix Eecondorder control systems
GregAmmar: TheQR algorithmfor orthogonalHessenberymatrices
William B. Gragg: Snbilizntionof the UHeR atgorithm
CORELINEARALGEBRA II
Fudren Zhang: Incqualitieson SchurComplements
K. Okubo: p-radius and sow matrices
TbtjanaPet*: spectrumand commutativirypresemingmappingson Hermitian
natrices
L. Elsner: o,thogorwl basestlwt leadto syrnrctic rwrmegativemarrices
RichardYarya:Hardy'sincEtality and uhramztric matrices
GeorgHeinig MinilruI rar* block Hanlcetmatrix efrensionsandpartial
realization
EIGENVALUES AND SINGT'LAR VALI'ES
Hector Rojo: On relaed boundsfor tlw extnemeeigenvalues
G. W. Stewart:On the eigenstrrctureof gradcd matrices
P6l R6zsa:Spectrarpropertiesof symmztricattyreciprocarmarrices
oscar Rojo: New lower (upper)boundsfor tr, smrrnest(rargest)singurarvaluc
Xing*i Atan: On tlv singutarvahrcsof matri.x,surlu
Chi-Kwong Li: Inequatitiesof tlw singularvaluesof an off-diagoml bbck
of a Hermitian matrix
voLKER MEHRMANN: Jordanand schurforms in lie and rodan Algebras
2:5U3:15pm
3:2f3:45 pm
INTERACTI\IE.TDff SESSION I
JohnR. wicks: student-centeredleaming usinga MathernaticaIrb
EugeneA. Herman:Thelinear algebramodnles
projea
2:5V3:15pm
3:2U3:45pm
MINISYMPOSIUMon LinearAlgebraMethodsin Statistics t: Organizedby
H. J.Werner
AugustynMarkiewicz: Matrix ircqualities andtltcir statisticatapplications
GeorgeP.H. Styan:Somecommenlson somesekued matrix incEtotities and equalities,
wilh somcstatisticalapplicuions
2:5f3:15pm
3:2f3:45 pm
MINISYMPoSIUM on Matrix F4uations,Inertia"Stability& srabilization t:
organizedbyB. N. Datta
PeterLancaster Optirnl decayof energyfor a systemof osciltators
T. Ando: Simultaneousstrongstability
page l8
Image: No, 20
Thursday,June4,lggg (cont'd)
2:5V3:05pm
3:1G3:25pm
3:3O-3:45
pm
NUMERICAL AND COMRMATIONAL LINEAR ALGEBRA
II
vlastimil Pttk: Kryrav seEuences
of matimor rengthand convergenceo! GMRES
Jinxi Ztao: A quasi'mhimal nsidual nuthod basedon bromplete orthogonali?prionfor
non-sytnnetrh liru ar systems
TlenEk Strako5:Accuracyof three-termutdtwo-term rccunences KryIov
space solvers
for
2:5O-3:05
pm
3:1O-3:25
pm
3:3O-3:45
pm
MAIRIX PROBLEMSFROM CONTROL AND SYSTEMSTHEORY I
A' c. M. Ran: comprete setsof sorutionsto algebraic Riccai equatiots
PeterBenner: Thediskfinction atd its relaion to algebraic Riccatiequarions
Me I. Garcfa-Planas:structural invafiantsfor lircar differentiat-algebraicequatiow
2:5V3:05pm
3:lO-3:25pm
3:3O-3:45
pm
COMBINAXORIAL MAIRIX TI{EORY I
PeterM. Gibson: combiratorially onlngoml matricesand relatedgraphs
Bryan L. Shader:Sparseonhogonalbasesfor hyperplancs
Arnolcl R. Kreute,r:An upperhouttdfor ttw pemtancntof a rcnwgotive matrix
2:5O-3:05
pm
3:lf3:25 pm
pm
3:3O-3:45
OPERAXORS
Luis verde-Star: one-sidediwerses of lincar generatheddifferentialoperators
Joan-JosepCliment: Generalconparison tlvoremsfor sptittingsof rwrwingularbowtdedoperators
SteveGalitsky: On comparisonof closeness
rrre(Nures
of positively4cfued operatorsnumericalexperiments
4:154:40 pm
4:45-5:l0 pm
INTERACTTVE-TDff SESSIONII
Elias Deeba: Teachinglinear algebra ushg an ilrteractivetefr
GeraldJ. Porter:Teachinglinearalgebraasalab course
4:154:40pm
4:45-5:10pm
MINISYMPOSIIM on LinearAlgebraMethotlsin StatisticsII: OrganizedbyH.J.Werner
N. Rao chaganty: The atulysis of longittdinal data using quasi-leastsquares
H. J. werner; on somcestimuhg techniquesfortlw generalGauss-Markavmodcl
4:154:40pm
4:45-5:10pm
MINISYMPOSIUM on Matrix Equations,Inertia,Stability& StabilizationUr:
OrganizedDyB. N. Datta
David Carlson:Recentadvancesin a pmbtcm on invariants
Leiba Rodman;Lircar matrixequariowand ruiorul matrixftutctions
4:154:30pm
4:354:50pm
4:55-5:10
pm
NT'MERICAL AND COMPUTATIONALLINEAR ALGEBRA III
Mei-Qin chen: The stabilityand convergence
of a direction set basedalgorithmfor
adaptiveleastsquaresproblems
Tibor Boros: Fastsolutionof llandermonfu-type
probkms
least-squares
KuiyuanLi: An algorithmfor thegeneraliudtridiagonaleigenvahrc
problem
4:154:30pm
4:354:50 pm
4:55-5:10
pm
MAIRX PROBLEMSFROM CONTROL AND SYSTEMSTHEORY II
F. hrerta: Topologicalstructureof tIE setof (A, B)t -invuiwrt subspaces
J. cloteu Boundingthedistmt of a controllable systemto ut uncontroltableone
Cristina Jord6n:Controllability completionsproblemsof matricesin upper catwnicalform
4:154:30pm
4:354:50pm
4:55-5:10
pm
COMBINAXORIAL MATRX TI{EORY IT
J. A. Dias da Silva: Rat* partition of a matroidaild the rar* partition of itsdual
JamesR. Weaver:Tburnanewmarices andttte Brwldi-Ij Conjecture
Olga Azenhas:Combimtorics of Linlewood-Riclardsonfiltingsand matrix spectralprcbtems
4:154:30pm
4:354:50 pm
pm
4:55-5:10
MAIRIX PERTUBATIONS
Xiao-wen chang: componentwiseperturbotionanatysesforthe eRfactorization
Erxiong liang: Penurbationsin the eigenvalucsof a symmetrictridiagonnl matrix
Yimin Wei: Perturbationbowtd of a singular linear system
5 : 1 5 - 6 : 1p5m
ILAS BusinessMeeting
Aprll 1998
poge l9
Friday, June5, 1998
8:30-9:15
am DANIEL HERSHKOWTTZ:The CombinatorialStructureof GeneratizedEigenspace
s
fum Nowtzguive Matricesto GeneralMatrices
9:2o-I0:05am NICHOLAS J. HIGHAM: QR Factoriution with CompletePivotingand AccurateComputatiowof tttc SVI
INTERACIIVE LAB SESSIONI:
10:35-ll:30 am Gerdd J. Porter:Lab Session
ll:35 am-I2:30pm EliasDeeba:Lab Session
10:35-1
1:00am
l1:05-11:30am
l1:35-12:00
noon
12:05-12:30
pm
l0:35-11:00am
10:05-1
1:30am
11:35-12:00
noon
12:05-12:30pm
10:35-1
1:00am
l1:05-l 1:30am
11:35-12:00
noon
12:05-12:30pm
MIMSYMPOSIUM on Linear Algebra Mettrodsin StatisticsIII: Organizedby H. J.Werner
Cul D. Meyer: Sensitivityof Markov chains
Adi Ben-Israelihkgration usingmatritvoluntc: Appticationsin probability and statistics
PeterMcCullagh: Facnrial modclsand iwariant subspaces
JohnS. Chipman:Lircar restrictions,rar* reduction,and binsedestifirationin linear regression
MINISYMPOSIUMon Matrix Equations,Inatia, Stability& Stabilizatiotllt: OrganizedbyB. N. Datta
Bryan E. Cun: Generaliutions of Sylvester\tlworemand tluir associued canonicalforms
Biswa N. Datt&:Incrti4 stability, and stabilization
P.A. Fubrmann:on thc Lyapunovequotionand coirwariantsubspacesof Hardy spaces
S' P.Bhattacharyya:A generalizationof ttrc classicalHermit-Bielcr thcoren with applicaionEto control
MINISYMPOSIUM on NumericalLinear Algebra t: orgaaizedDyM. overton
JamesW. Demmel:WlwncantheSVDbecomputedaccurarely?
Danny Sorenson:Deflation schcmesof implhitly tesnrted Arrcldi mcthods
Ming Gu: On rar*-revealingfactorizations
Alan Edelman:Algorithmsfor nearestcanonicalforms
STRUCTUREDMAIRICES
10:35-10:50
am GeorgHeinig: snbilized superfastToeplitzsolversbasedon interpolation
l0:55-l 1:10am stevenJ. Kifowifi A superJastversiono|tlv sptitschur
algorithm
1 l : 1 5 - ll : 3 0a m David T. Clyctesdale:A superfastalgorithmfor solvingpositive
dcfnite bbck Toeplitzsystems
11:35-ll:50am william F. Tfench:Asynptotic distribution of tltc even
and odd spectraof real synmeyic Tbeplitzmatrices
I l:55-12:10pm Niloufer Mackey: Structure'preservingJacobimethods
for doubly-strrcturedmatrices
12:15-12:30pm vatlim olshevsky: Eigenvectorconputations almost-unitary-Hessenberg
for
matrices
via discretetransmissionlines
l0:35-10:50am
10:55-1
1:l0 am
1 1 : 1 5 -l1: 3 0a m
11:35-1
l:50am
11:55-12:
10pm
12:15-12:30
pm
2:O0-l2:25
pm
2:3V2:55pm
3:354:30pm
4:35-5:30pm
NONNEGATIVEMAIRICES
JaneM. Day: On a tlteoremand conjectureof Gartoff
ShaunM. Fallat: On the spectrumof totaily nonnegaivenntrices
Roy A. Mathias: A newapproachto totally positivematrices
Naomi Shaked-Mondercr:support-restrictedrar* I representotions
of completetypositivematrices
D. Dale olesky: Relationsbetweenpermn-Frubeniusresults natrixpencils
for
penon-Frobeniusthcory
SiegfriedM. Rump: Generalized
JIJDITH J. McDONALD: Splittingsof SingularM-natrices
FERNANDO c. sILVA: onttw Eigenvaluesof sumsandprodrcts of Matrices
INTERACTIVE LAB SESSIONII
EugeneA. Herman:Lab Session
JohnR. Wicks: lab Session
poge 20
lmagte: No. 20
Friday,June5, l99t (cont'd)
MINIsYMPosil.tM on structuredMatrix problemsrr: organizedDyLothar Reichel
3:25-3:50pm vadim olshevsky: Pivotingfor stracturedmatriceswith applicuions
3:55-4:20pm Thilo Penzl:A cyclic low rat* smithmethodfor large, sparseLyapurcv equatioru
4:254:50 pm Misha Kilmer: Preconditiottcrsfor structuredmatrices arisingin milw dctection
4:55-5:20pm JamesG. Nagy: Kroruckcr prodrct decompositionof block roeptitz matrices
3:25-3:50pm
3:55-4:20pm
4:254:50 pm
MINISY-MPOSIUMon Matrix Equations,Inertia Stability & Srabitizarionty: OrganizedDyB. N. Datta
ShaulGutrnur: Worstcasedesignfor robustcompensation
GeorgHeinig: Inertia of n atrix poWmials and Block HafueI mrilrices
SergioBittanti: Invariant reformulatioruof periodic systems
3:25-3:40pm
3:454:00 pm
4:O54:20pm
4:254:40 pm
4:45-5:00pm
5:05-5:20pm
5:25-5:40pm
5:454:00 pm
CORE LINEAR ALGEBRA IU
Michael Neumann:Partial norns and thc convetgenceof gmeral pro&rcts of flntrices
RogerA. Hon: Themoncnt mrtrir and someapplicuions
Vlaclimir A. Strauss:Theilratrix,momcnprcblcn: a dcfuitizabte version
PedroFreitasi On an action of tlw symplecticgroup
KennethR. Driessel: Ontlw geometryof someisospectratsurfaces
S. P.Thn: Prbrcipal congruencesubgroupsof tlw Heckegrcup
Natdlia Bebiano:Somematrix ircqualities in physics
JohnMaroults: Ontlu numericalrangeof ruiorwl matrixfinctions
3:25-3:40pm
3:454:00 pm
4:054:20 pm
4:254:40 pm
4:45-5:00pm
5:05-5:20pm
5:25-5:40pm
5:45-6:00pm
NUMERICAL AND COMPTIII{TIONAL LINEAR ALGEBRA IV
David Saunders:Solvingparametic linear systens
Bany Nelson: How solving linear equationscan bencfitfrom differewial equuion
YoopyoHong: A group hotrwlogymcthodfor symmetriceigenproblems
EugeneC. Boman: Computingpreconditionersvia subspaceprojeaion
FrangoiseTtsseur:Strucured baclortarderrcr and cottditionnunber of quadraticeigenvalrcproblems
Xiao-WenChang: Componentwise
pertubationatwlysesfor thc eRfactorization
ThomasSzirtes:A tnatrix nnthod to generatedimensionlessvariables
FrankUhlig: Gewral polytomial rootsaildtlair multiplicitieson O(n) nennry and O(nz) time
COMBINAXORIAL MAIRIX THEORY III
3:25-3:40pm Elena Shamis:On a duality betweentwtices andD-proximitics
3:454:00pm P.van denDriessche:Thepower methd in Max Algebra distarcefour
4:054:20pm Michael Tbatsomeros:Exnennl propertiesof ray-rcnshgular mttrices
4:254:40 pm Carolyn Eschenbach:Eigenvalw distribution of certain ray patterns
4:45-5:00pm A. Tbrres-Ch6zuo:Constructionof binary const(nt weightcodeswith minimumdistance4
5:05-5:20pm G. Sampatb:Orderedadjacencymatrices
5:25-5:40pm M. Antonia Duffner: Liwar transformdionsconvertingimwnants
5:45-6:00pm Ant6nio Leal Duarte: Eigenvaluesof principal subm.atrices
of acyclic matrices
pm
6:30-10:00
BANQIJET: GreatHall, Memorial Union
Saturday,June 6, 1998
8:3O-8:55am
9:0O-9:25am
9:3G9:55 am
10:00-10:25am
MINISYMPOSIUM on NumericalLinear Algebra ll OrganizedDyM. Overton
Roy Mathias;Carefularwlysiscan increasespeedof eigenvaluealgorithms
Julio Moro: Weyl-typerelalive pertubafionbottndsfor eigenvaluesof herrnitiannatrices
NicholasJ. Higham:Thenearestdefnitepair ailtltc iuwr numericalradius
Ilse C. F. Ipsen:If a natrix has only a singleeigenvalue,how sensitiveis this eigenvalue?
poge 2l
April 1998
Saturday,June 6' 1998(cont'd)
8:30-8:45am
8:50-9:05am
9:10-9:25am
9:30-9:45am
9:5O-10:05
am
10:1O-10:25
am
CORELINEAR ALGEBRA ry
Andr6 Klein: On tlw sol.utianof Stein'seEtUion theBezoutian,and stAiSical it{onnaion
Tian-GangI*i: Positivesemi-defnitencssin a gercraliud group algebra
RafaelBru: Gmup involunry fiatrices
M. C. Gouveia:Ontlw SteinandLyapwnv equatioruover rings
RalphDeMarr: Grcupsof matrices
Ilya M. Spitkovsky:An updateon tlw facnrization problemfor block triangular matrixfwctions
8:30-8:45am
8:50-9:05am
9:10-9:25anr
9:30-9:45am
9:50-10:05
am
l0:1f10:25 am
MATRIX PROBLEMSFROM CONTROL AND SYSTEMSTHEORY III
A. Compta: Obsemablepairs havingprescribcdobsemableandnilpotent suppkmennry blocks
K.-H. F0rster: On tlw tnnncgative reati?ntionproblcmfor ratioml Irmctions
nethod
FernandoRincon: Tbwardsa designof a rcbrut eigenvahu assigttlrr,ent
geonetric
approach
McD. Magrer Lhcar difrercntial-algebraicequations.A
NdstorThome: Balancednndcl reductionof discrcte-timerystemsby stue-feedbacks
SusanFurtado: Completionproblcns and someapplhaionsto linear systems
8:30-8:45am
8:50-9:05am
9:lO-9:25am
9:30-9:45am
9:5f10:05 am
l0: lO-10:25arrt
COMBINAf,ORIAL MAIRX THEORY IV
Sang-Gul*e: On a sigt-panernmatrix and its relatedalgorithmsfor L-rutrices
Frank Hall: Signpatternsof idenpotentmalrices
ZhongshanLi: Potentially nilpotent signpattern natrices
David P.Stanford:Pafiernstlw allow givenrow andcolwnnsums
Jeffiey Stuarl Signk-poten signpattern matrices
JuanTorregrosa:The sign symmetricP-nutrix completionproblpm
8:3O-8:45
am
8:50-9:05am
9:10-9:25
am
9:30-9:45am
9:50-10:05
am
10:lfl0:25 am
MATRIX CANONICAL FORMS
DennisI. Merino: A noteon quaternionmurices and complcxpartitiorrednatrices
D. StevenMackey: Haniltonian squarcrootsof slcew-Hanikoniann atices
Me I. Garcfa-Planas:
A wrmal form for contragradientpercils dependingon paratneters
Bit-ShunTam: On tlv Jordancatnnical forms of an eventuol$ nonneguivemt trix and relatedtopics
DennisI. Merino: TheJordancanonicalformsof orthogotul and skew-symmetricmatrices
Mark A. Mills: Layersof marices: Theconjunctivity-similarityequivalerrceclassof matrices
11:0f 11:45
am URIEL C. ROTHBLIJM: Marix Scalings:Existence,Clwracterization,and Computation
NumericolAnolysls
ond Computers:50Yeorsof progress
University
of Monchester,
Monchester,
Englond,Junel6-17, lgga
This conferencgorganizedby Nick HighamandDavid Silvesterof theManchesterCentrefor Computational
Mathematics,
ties in with the celebrationsin Manchesterto mark the 50th Anniversaryof the birth of "the Baby", the first stored-program
electronicdigital computer-born at theUniversity of Manchesteron June2L, 1948.The meetingruns from 9 amon T\resday,
June16until 2 pm on Weclnesday,
June17.Theaim is to describehow numericalanalysishasbeeninfluencedby thedevelopment of computersov€r the last 50 yean. The meetingcomprisesinvited talls coveringkey areasincluding numericallinear
algebra,optimisation,ODEs, PDEs anclcomputationalfluid <tynamics,and will containhistorical remarks,perspectivesand
anecdotes.The influenceof high-perfonnance
computingon tbe funue of numericalanalysiswill alsobe assessed.
A dinner will be held on the Thesdayevening.The speakersinclude:JackDongana(Knoxville), Brian Ford (NAG), Ian Gladwell
(SMU,Dallas),GeneGolub (Stanford),CleveMoler (TheMathWorks),Bill Morton (BathandOxford),Mike Powell(Cambridge),Mark Sofroniou(WolftamResearch),
AndrewStuart(Stanford),Nick Trefethen(Oxforcl),andJoanWalsh(Manchester).Forfurtherdetailsandregistrationformspleasevisithrrp:,/,/www.ma.man.ac.uk/NAC98.
Thereisnoregistration
fee for graduatestudents.The meetingis sponsoredby the LondonMathematicalSociety,NAG Ltd., andWolfram Research.
page 22
lmoge: No. 20
WesternConodo LineorAlgebro Meeting
.|,
Victorio,British
Columbio,July30-3 1998
The WesternCanadaLinear Algebra Meeting (W-CLAM) providesan opportunity for mathematiciansin westernCanada
working in linearalgebraandrelatedfieldsto meet,presentaccountsof their recentresearch,andto haveinformal discussions.
While themeetinghasaregionalbase,it alsoattractspeoplefrom outsidethegeographicalarea.Participationis opento anyone
who is interested
in attendingor speakingat Oe meeting.Previous}V-CLAMswereheldin Regina(193), Letrbridge(lD5)
andKananaskis(1996).
The fourthW-CLAM will be heldin Victoria,8.C.,July 3G31, 1998.Participants
areinviled to contributea talk on linear
algebraor its applications(3G45 minutes,dependingon the numberof participants).Abstractsshouldbe sentto SteveKirklandkirklandGmattr.uregina. ca orMichaelTbatsomeros
tsat6math.uregina. ca byJune15,1998.Theparticipation
fee for the meetingis $20 (Canadian),to be collect€dat the meeting.This feewill be waived for participatingstudents.Some
supporttowardsexpenseswill be availableto a maximumof 6 studentsor post-doctoralfellows who wish to give a talk, on a
first+omefirst-servedbasis.For up-todateinformationon W-CLAlvt'98 pleasevisiu http : / /www.rnath.uregina. cal
-tsat/wclanr.html/.
TheW-CLAMOrganizingCommitteecomfises:
HadiKharaghani:
hadigcs.uleth.ca; Stephen
Kirkland: kirkland0math . uregina . ca; PeterLancaster:lancaste0acs . ucalgary. ca; DaleOlesky:doleskl€csr .
csc . uvic . ca; MichaelTbatsomeros:
tsatGmath. uregina . cai andPaulinevandenDriessche:pvdd0nrath. uvic . ca.
'98,
Following W-CLAM
the Univosity of lrthbridge will be offoing a mini-courseentitled'Coding, Cryptographyan<l
ComputerSecurity''on August 3-7, 1998.For further informationon the mini-coruse,pleasecontactHadi Kharaghanior for
up-todateinformationvisit hetp . / /tttwtr.cs .u1eth. ca/workshop/.
SeventhInternotionolWorkshopon Motricesond Stotistics,
in Celebrotionof T. W. Anderson'sSOthBirthdoy
FortLouderdole,Florido:December I I -l 4, 1998
The SeventhInternationalWorkshopon MatricesandStatistics,in celebrationof T. }V. Anderson's80thbirthclay,will be hel<l
University, Fort Lauderdale,Florida, on the weekendftom Friday,December1l throughMorulay'De'
at Nova Southeastern
cember14, 1998.For their supporrof this Workshopwe arev€ry gratefulto the InternationalLinear Algebra Society(ILAS),
University, urd the StatisticalSocietyof Canada(SSC).The InternationalOrganizingCommitteefor this
Nova Southeast€rn
WorkshopcomprisesR. William Farebrother(Victoria University of Manchester,England), Simo hrntanen (Universi$ of
Tarrpere,Tampere,Finland), GeorgeP. H. Styan(McGitl University,Montrdal, Quebec,Canada;chair), and HansJoachim
University
Werner(University of Bonn, Bonn,Germany;vice+hair). TheLocal OrganizingCommitteeat Nova Southeastern
is
seventhin
(chair).
the
This
Workshop
Zrang
Fuzhen
and
comprisesNaomiD'Alessio,William D. Hammack,David Simon
(2)
New
Zealand:
Auckland,
1990,
August
(l)
Finland:
follows:
T[mpere,
u seri"s.Theprevioussix Worlshopswereheldas
July
(5)
England:
Shrewsbury,
July
1995,
(4)
Canada:
Decemberlgg2, (3) Tartu,Estonia:May 1994, Montr€al,Qu€bec,
1999.
August
6-9'
Finland,
in
Tbmpere,
hetd
will
be
Workshop
1996,and(6) istanbul,Tiskey: August 1997.The Sttr
The purposeof this SeventhWorkshop,thefirst to be held in theUnited States,is to stimulateresearchand,in aninformal
in theinterfacebetweenmarix theoryandstatistics.This Worlshop will provide
setting,to fostertheinteractionof researchers
in
the
areasof linear algebra/matrixtheoryand/orstatisticsmay be betterinformed
working
scientists
which
a forum through
andmay exchangeideaswith researchersftom severaldifferent countries.
newest
techniques
and
of the latestdevelopments
will be publishedin a SpecialIssueon Linear Algebra and Statisticsof
papers
Workshop
from
this
Selectedfully-refereed
will includeinvited andconributed talls. Thesetalks andthe informal
The
Workshop
Applications.
lts
Linear Algebra and
of ideas.We expectspecialcoverageof someof the topics that
exchange
guarantee
an
intensive
wili
workshopatmosphere
havehighlighteAfeO Anderson'sresearch:muttivariatestatisticalanalysisand econometrics,aswell asmatrix inequalities.
Accommodation,at very favorablerates,has beenarrurgedat the Rolling Hills Hotel & Golf Resort(Official Hotel & Resort of theMiami Dolphins,film locationfor the hit cornedy"caddyshacK');tbis hotetis aboutl0 minuteswalk from Nova
University.Pleasevisit hrrp : / /www.polaris . nova . edu/MST/conf /FMl{/, whereall availableinformation
Southeastern
aboutthis Workshopwill be updateclfrequently.
April 1998
poge 23
ImageProblemCorner
We presentsolutionsto Problems1t-4, l9-1, l9-2 and19-5andwe introducesix new problems(page32). We invite readers
to submitsolutions,aswell asnew problems,for publicationin Image.We look forwardparticululy to receivingsolutionsto
hoblems l8- l , l9-3 and 194 ! Pleasesendall materialto GeorgeStyanBOTH in Ulg codeembeddedastext only by e-mail
to sEvanoMath.McGill . ca AND by regularair mail (nicelyprintedcopy,pleasel)to Professor
Creorge
P.H. Styan,Frlitorin-Chief: Image,Dept. of Mathematics& Statistics,McGill University, 805 ouest,rue SherbrookeStreetWest,Montreal,
Qu6bec,CanatlaH3A 2K6. Many thanksgo to lrurge SeniorAssociateEditor HansJoachimWernerfor his tremendoushelp
with this Imagekoblem Corner.-Erl.
Problem 1&1: 5 x 5 Complcx lladenard Metriccs
Proposedby s. lv. DRURI McGilt (Iniversity,Montrcal,
Qucbec,cailada.
TheHitor has rct yet receiveda solution to thisproblcm-indeedeventlw proposerhas
wt yetfound a solution!
show that every5 x 5 matrix t/ witb complexentriesui,r of constantabsolutevalue
one that satisfies(J*[J = 5r canbe
'|,yA asthemaEix(rt r)i,* wherer.ris a complexprinriiivenftr root of unity
by applyingsomesequerceof thefollowing:
(l) A reanangement
of therows, (2) A reanangement
sf th6selrmns,(3) Multiplicationof a row by a complexnumbs of
absolutevalue one,(4) Multiplication of a columnby a complexnumberof
absolutevalue one.
hobhn 1E4: Bounds for a Ratio of Metrix Tbaccs
Proposeclby SuuaNczHE Lru, (Iniversitdt Basel,BaseLSwitzertand.
L€tD > 0beann x nmatixwitheigenvaluesll
)...)
lr,xbeann
that
I
t(x,Ex;2
i
x ftmatrixsuchthat
x,X = I andn) 2&.show
Ef=r(Ar* .l'-i+r)z
4Ef=r)r ),,-, +1
solution 1t'f'1 by NarAln BEBIANoandJoAo on
Pnovron Nr:,A,universityof coimbra, coimbra,ponugat
&
CHI-Kworirc Lt, Collegeof WtiamandMary, Williansburg,Vrginiq
USA.
We prove the inequalities for complex Hermitian ma'ices.
L e t X b e n x /csuchthat X* X = I1r. Thenthereis a
unitary
matrix u with X in the first /c columns. Let M :=
[J*E{J =
with A x*Ex. Thentheinequalities
(!. C1)
\^B)
,.
become:
tt (A2.
*_pB*). Di=r(ti * ,\"_i*r)z
trA'
aDf=,litr,_j+r
The first inequalityis clear' Tb provethe secondinequality,
we focuson the optimal matrix M rhatyieldsthemaximumratio
tr (A2+ BB.)/tr A2.
Wefirstconsiderthecase_when
n =2k. SupposeBhas
singularvaluedecompositionByEz,wherey
f x / c u n i t a r y m a t r i c e s , a n c 6 - d i a g ( D 1 , . ' . , D i ' w i o a t 2 ' . - . - i l - ) 0 . w e m a y a c r u a l l y=a s s u m e t h a t g = -andZ
6 , o rue
herwise,
/
wtw
t ot (
;iY;.
::o]i'"
matrix
;!ttt).
Now,ror everyHerrnitan
s, andr € R in a neighborhood
or 0, wecanconsrrucr
a
( e1l B(,) \ _
-_d,s114"its
= e-"o
'
ci'| )
\B(i)'
whue/(t) is & x [. Moreover,
thedifferentia!]1nrncti91/q)= t (A(t^)2+ B(t)B(t).)/h A(t)z
atrains
a maximum
at
r = 0' LetQ = Ix @0,,-r.Then,tr (A2+ BB') =t (Mleiurotr'1,iij =;i;o-dir,fj.ariropr"compurationshows
wge 24
lmage,'No.20
that
(slyI:{!t)
+2f'!sY*ry)
=Q
u1U@
dl
-f
"fr=o =
#r"tr(t)l
atr
ft(M2e)
Since,Sis arbiEary,we have
[Iur,,el
ft (Az * BB*)
(o
_ .)lM,QIuIQl
ft (A2)
_B*o ; " " . )
BC
+
\/B
b (Az+ BB*)
-B*/\
-2
(:,
o)
-0.
tr (.42)
onecanthendeduce
thatAB = rBC rorz =_lrli e\/tr(*-t
> 0. Thus.4
isin blockformr{r o . ..@A^
according
totbemultiplicities
of thedistinctsingulaivaluesoi B. !_BB')
Now,wecanuseaunitarymatrixlz of the fotmw =
Wt "'@W^
@
conformedwith tbatof .t{ so that-W'!W-:
$.g(or, ....,or). Clearly,al"n." W*BW = g. Replace
Mby(W* aw.)M(wPw),wernayassunethata =diag(ai...,o1)
andB=diag(61,...,br)arebothin<tiagonal
form' andwe still have-r{B - TBC - Hencec = diag(cr . . . J-l
ir
e
rf
<i
nrs - posrtiue'liigulr, ,,rt*.. Iu prticular, M
, ,
isadirectsumofdiag(a',^+lrt...,cr)ge ula2 x 2principalsubmatrices = (?i 6i
rcesM;
U,
f i - 1, " ., rn. Moreover,
(. u, ir ) ^t
)
kk
b (A2 + BB*)/tr A2 =I@?+b?)/Do?.
f=1
For j - 1,"',tu,usingthe tactnat (!.
i=l
1:) isunitarilysimilartodtag(p5,v)forrwoeigenvaluespi
andvi af M,
w€mav-v/rite
oi\ti *"SYir,riilorrnor, ,, *o,*st),oneshouldclearlychooseci
robetbesmallest
:j+b? eigenvaluesofMavailableafterremovin8pr,...,pm,andzl,...,um.Consequently,todeterminetbemaximum
ft(A2*
B B') f tr A2, onemayconsiderthemaxiumof
\rm
- - ur!r(o;(p; + !) p;v;)* Df=-+r 4;
"amongallpossiblechoices
of rn+&eigenvaluesFr,...tpmtt/Lt...tumt1m*Lt...,qxofl,where0
1m, 1&,subjectto
the constraintsthat
Now,maximixng
s undertheabove.ooro*1,:ll't|il;
,;i;";-
upperbo'ncrfor oe optimar
the
s byrela:ring
conditionson ai for i =
], . . . ,rn, namely,we only assunethatal + b? = ai(pi * v) - pivi for someeigenvalues
a; €fv;,ttiJto, j 3-m,inaa; tobeaneigenvAire-forl
Itt,-..,Ph,ut,-..,t/k of Dwithoutrequiring
> -. thus,we
maximize
D!=r@;(p;*r) - t';r;)
D!=r"?
For a fixed choiceof p; and vi, we choose(or, . . . , a1) to ma:rimizethe expressionandobtaintbe equations
0t; + v;1
-zFp a;
J,
Li=L ai
-t)
Df= L(ai6i + ri - tttri)
i
-1,...,k,
at a critical point. This leadsto
srlc
Li=Lai
,
zilo(rc,i1ti +r)-
ttiui)
r. e t . a = a ; / ( n + zr).T h e n zc=;$ fu w h er eSr = Df=r ftr r .+ vi\2and,gz=Df=,
Fivi.T\usr r = 2s2f S1.Inother
words,the maximumoccursat
If-t = Jpi + r)2
^1 _ e
_
_ 1+
2a
a*=L
piui
Df=tfui-'il'
rs&
+ Li_t
ltjvj
'
April 1998
poge 25
Next, onecanprovethat this is no largerthan
Dt=r(),- ),,-i+)'
4Dl=, lr)n-j*l
In fact, if we arrange p; d ui in descending order c1
Dr'lt("i -rrr-i+r)' > Diir("i -x;;)2 foranypermutation(ir,...,i4)ot(1,...,2/c).
Hence,wernayassumetbat
qwhenmaximumoccurs.
Fr 2 "')
jwittrl S j S ksuchthat
Itx )- vp ) "'2
Moreover,if
thereexists
1ti,r) * (\i,ln-i+r), onecanimprovethemaxmumby replacing(pi,r) by (li, f"-t*r) at thefust occuenceof this
inequality.
For thecasewhen n > 2&,we can apply the argumentsto the leading2k x 2k matrix of the optimal ma6ix M andget the
conclusion.
tr
solution lt-42 by snurc,xczge Ll.:u,(JniversitdtBasel,Basel,switzertand.
It's easyto seethefirst relationship.Define
1I
1
,
_
4=
t1X'X _ t)
ilosg
,toSh
,tr
whereg = tr X'82X, h = tr (X'EX)2 andLt = .t. Clearly,
dg= !6 y'82dx -
|r, x,xx x,Edx ft LX,dx.
By lettingdd = 0 andsolving.L we get
! y ' 2 2 - ? x , D X X , E= l r , E z X X , ?
)rX,
g
h
g
h '( X , E X
(1)
!x' sx -ix,Exx,Ezx- Lx,Dzxx,EX
-i(x,EX),
(2)
Lx,x"x-|x,>rxx,Ex - Lx,>xx,zrx-l{x,>x)".
(3)
Postmuldplying
ir by EX gives
Takingtransposeresultsin
From(2)and(3)weseethat(i-fl(uN-NM)
= 0,andhence
MN - NM,whereM= x,D2X > 0,ir =X,EX ) 0,
=T(JT:,N=TVT,,
:a_-i-i:0?s2t->_Ztr>ir>0.Withoutlossofgenerality,assumeE=diag(lr,...,,\,).WriteM
T ' X ' = W , w h e r c U= d i a g ( u r , . . . , u>&0) ,y = d i a g ( u r , . . . , u r )
) 0 , T t T= T T t = 1 .T h e n ( l ) l e a d s t o
i*r'-|vwz=Luw-1n,*
(4)
Examine(rdt , ...,rrdn),thei-th row of W (i = I ...,&) andreunite(4)
,
as
uiiF(j)
= 0,
(5)
w.lTrfi)=ili-,Tr,\i,i":j.*r?,i=1,...,n.It'snorpossiblerhdtw;1-...=ttin=0,asww,-1.D,ueto(
w e m u s t h a v e e i t h e r ( a ) w ; , 1 0 , F=(0r;)o r ( b ) 0 # w ; r * u i s l 0 , F ( r ) = F ( s )
=.0,r-fs.In(a)wehave
poge 26
lmage: No.2O
i'e" h2u; - (2ch - c2)a?'theng = A (as
ui > v?),whichleadsto thelower bound(first
relationship),not anupperbound.
In (b) we have
)"+ )" ='frr,)"),=
Solving u; and u; w€ obtain
-g=- If=r'i
h
Ef=r ui
ft - u;.
- @Ef=r()" * ).,)2- Ef=r)"),
(*)'Ef=r()"
+ \r
and hence
(6)
Notethatrands(1s.r *s3.n)relaretof (f =1,...,&).Maximizing(6)weobtaintheresult.
tr
coraMexr' Note that theresulthasapplicationson efficiency
comparisonsin statistics(partly presentedat TinbergenInstitute,
Amsterdamin 1993andAdam Mickiewiczuniversity,poznan
in 1995).
hoblem 19.1: Eigcnvalues
Proposedby JtincrN GR.B & Gorz T*BNKLER,(Jniversittfi
Dortnund,Dortmun| Gemuny.
Let a andb be two nonzeron x 1 rearvectorsandconsiderthe
mahix
A = oaa, + F
whereo,F,.1,6 arerealscalarssuchthat
F=-tand72
* 7ba, + dbbr,
"b'
=_o6.Findthe(nonzero)eigenvaluesof
A.
solution 19'1'1 by DAVID A. Hanvu-tE, IBM ThonusJ. watsonResearch
ceneq,yorfuownHeigttts, Newyork, usA.
L&t c, y, ra,andz representrealnumberssuchthatEu)= e, oz = _1,
UU = .1,yz = 6,so that
A = oaa' - 7ab, + fba/ + 6bb, = (ca* yb)(ura* zb),.
Thatsuchnumbersexistis evidentuponobserving(in light of theconditioo.l" - -o6,
which implieseirherthato and6 are
opposit€insignorthatonflbottrof
themequal0)that(l)ifo > 0(inwhichcase6< 0),wecanrakeu- a {d,
z = -y, andeitherv = r/-6 ory = -t/=6 (gependingonwhetherl20orr
< 0); (l)tr'" = 0(inwhichcase.r =0),
wecantakeu = I = 0andeitherz = u = t/6 or z = -y andu =
) 0or6 < O);and(3)
rlt'(Cependingon-ry-nenerd
- -a,x =
if o ( 0(inwhichcased) 0),wecantakero
lfr,urdeithei ; _V: -J6or, = y =.f tO"#"Orog""
whether7 ) 0 or r < 0). Then,it follows ftom a well-knownresult,which is givenby Theorem
2l.l0.l ornarvitte (1997)
f'Murix Algebra From a Statisti.cian's
Perspective",Springa-Verlag,New yorkl, thai a - 1 of the eigenvaluesof A are 0,
andtheremainingeigenvalueis
(ura* zb),(aa+ yb) - oa,a*7"7b- 7b,a+ 6b,b = aa,a+ 6brl.
tr
Solution l9-12by CHt-Kwot'tc Ll andRoY MATHTAs,TheCollegeof William& Mary,wiltiamsburg,virginia,
USA.
NotethatA = (a
b) B (o
D)', where
n=("
\r
P'\=("
bl
\r
'?\
6)
hasdeterminant
zeroby thegivencondition.Thus.4 hasat mostrankoneandthenonzeroeigenvalue(if exiss) equalstr A =
\r'v'r'uw"ErgsJUr
tr((a
b)B(a
A)')=tr(B(a
b)'(a
b))=a(ata)+6(D,r).
tr
solution 19-1.3by HnNs JolcHrM WERNER,universitdtBonn,Boru, Gernany.
It sufficesto considerthe following two exhaustivecases.
caseI' f,etc = 0' Then7 =0andB = 0sothatr{becomes
A= 6bb'whichisamatrixwithrank<l. Thismatrixhas
rank I if andonly if 60 I 0. If it hasrank l, thenits only nonzeroeigenvalueis givenby tr(/) = 6b,b.
April 1998
poge 27
- Bb)(" *
Case2. t-eta + 0. Then ir is easyto seethat.Acanbe rewrittenas.A = (aa
*r) which againis obviouslya
is given
matrixof rank S I . This matrixhasrank I if andonly if ac + +Bb. If it hasrank I , thentheonly nonzeroeigenvalue
/
^ \,
az
o
by tr(/) = (o + (o) {"" - Fb)= aa'o - ffb'b.
USA;KnzvszroF KRAMadisort,Wisconsin,
SolutionswerealsoreceivedfromDAVIDCILLAN, Universityof Wisconsin,
Poland;SreveN J. LEoN, UniversityofMassachusettssusxl & Mensr BrALKowsK\ WarsawUniversityofTbchrwlogy,Warsaw,
(ISAi
Louisiatw Universiry,Harnmond,
MERINo,
Southeastem
DENNIsI.
Dartnouth, North Dartmouth,Massachusetts,
ComellUniversity,Ithaca,
StanfordUniversity,SturfordCA,USA;Sttlvt-ER.SEARLE,
lnuisiatw,USA;INGRAMOLKTN,
YoNccETrtN,
ConcordiaUniversity,
NewYorlc,
USA;DerNa SHasHl, Tel-HaiRodmanCollege,UpperGalilce,Israeli
Montr1al,Qudbec,Canafu;WLLTAM F. TRENcH,Divide,Colorado,USA;HsNnv WoLxowIcz, Universityof Waterloo,
JiincBN GRoB& Gorz TnexrLak, UniversitdtDortnund,Dortmund,Germany.
Ontario,Canada;andfrom theproposers
Problem 19-2: Non-negadveDefinite Metriccs
Proposedby HlNs Joecxtrr WERNER,UniversitdtBorut,Bonn,Germuy.
) 0forallvectorss eCn. Provethat/isnon-negative
Amatrix AeAnxn issaidtobenon-negativedefiniteifRe(o*.,{c)
inverseAt is non-negative
definite.
definiteif andonly if its Moore-Penrose
Solution l9-2.1by R. B . BAPAT,Itdian Statisthal Instiute,NewDelhi, htdia.
(i.e.,
Since.Re(c*At) - lx'(A * A')s, A e C"xn is non-negative
definiteif andonly if A +,4* is positivesemidefinite
,'(A+A')t20forallxeC"'".)Suppose.4isnon-negativedefiniteandletxeN(A),thenullspaceof.4.Then
,*(A+ A')x = 0 andsince
A+ A* ispositivesemidefinite,
(A+ A.)t = 0.ThusA*o = 0andc e N(A.). Similairlywe
canshowthatc € N(A') impliest e N(A) andhenceN(A) = N(A.).II followsthatC(,4)- C(A.), whereC(.) deDores
-AtXfot
columnspace.This,togetherwiththewell-knownfactthatC(,4t)=C(r{.)leadstoC(11.1=C(At).Thus.r{t*
somematrix X. Then
AtAAt'- AtAAtX - AIX = Al,
andhenceAt = At A' At*. Using theseequationswe get Al + At* = At (A + A*)A1' , which is positivesemidefinite.
Therefone.Alis non-negative
dennirc.This provesthe "only if'part. The "if'part follows at oncesince.Att = .,{.
n
Solution l9A2 by CHt-KwoNc Lt andRoY MATHIAs, TheCollegeof William&.Mary, Williansburg,Virginia,USA.
Let S be invertible,andlet ft be a positiveinteger.Thena given.4 is non-negative
definiteif andonly if oneof thefollowing
holds:(a) S*.4S is non-negative
definite,(b),A- +.4 is non-negative
dennie, (c) .4 O 0p is non-negative
definile.Also, (d)
if ,4 is invertibleandnon-negative
definitethenAt = A-r - n-t a*(A-t;* is alsonon-negative
definite.Finally,since
A = (At)t it is sufficientto showthatA is non-negative
definiteimpliesthat /t is non-negative
definite.
SolutionI (via the Schur Decomposition):I.et A = UTU* whereU is unitaryancl? uppertriurgularwith theeigenvaluesof,4onthediagonal.
Letusassumethat,4has
/cnon-zeroeigenvaluesandthatt;;
f 0,fori ( /curdt;i = 0 fori > e+1.
By(a) above?isnon-negativedennite,
andsoby(b)sois?*?.. Since(?* T');; = 0 fori > & itfollowsthatthewhole
of theithrow andtheithcolumn
of (?+?') is0fori > i. SinceTisuppertriangularitfollows
that? = ?r O0,r_r where
definiteby (d) and(c), andso is .At = UTI U' .
fi is invertible.HenceTt = Tlt O 0,-r is non-negative
Solution tr (via SVD): Suppose.,4,is non-negative
definiteandhassingularvaluedecomposition
U DV. "I\en DVU =
U*AU andDVU +U*V'D isnon-negativedefinite.If
,4hasrank&,thenthelast(n-&)rows
otDVU arer&ro,andthus
tbelower(n - e) x (n - &) blockof DVU + U'V'D is zero.SinceDVII + U'V'D is hermitiannon-negative
definire,
its(1,2)and(2,1)blockmustbezaroaswell.
ItfollowstbatDVU = B@0,-p,whef,eBis&x Iinvertiblenon-negative
definite.Hence(J'V' Dl = B-r @0n-* is non-negative
tr
definiteby (d) and(c), andsois ,4t = V'Dl (J*.
Solution 19-2.3byHats JoacgrM WERNER,UniversitdtBonn,Bonn,Germany.
poge 28
lmoge: No. 20
SinceRe(r* A c) = x* (A + A')x 12,it is evidentthat.r{is non-negative
definiteif urd only if theHermitianmatrixA + A*
is non-negative
definite. Clearly,if B is Hermitiannon-negative
definite,thenx* Bo = 0 +=+ Bx = g. We recall that
a marix C is an EP-matrixif and only it N(C) = N(C*) or, equivalen0y,R(C) = R(C*); here,A/(.)denotesthe null
spaceand R,(.) the range(column space).We proceedas follows. First, we show that eachnon-negativedefinite matrix .A
is necessarilyan EP-matrix. With this in minG we then prove that .Al is non-negativedefinite whenever.,{ is so. For that
pupose, let z4be non-negativedefinite. Moreover,let c be an arbitrary vector from N(A). Then Ao = 0 and it follows
Invirtueof
from0=Re(r*, a)=x*(A*A')xl2that(.2{*A*)c=0andsoz{*c=0.ThusN(A)gN(A.).
AAt - At A; for noticethatA.At
rank(rA) - rank(.A*), therefore.A/(A) = N(A'), i. e. A is ER asclaimed.Consequently
t}e orthogonalprojectoronto?(.,{'). The defining
represents
the ortbogonalprojectorontoR.(r4)whereasr{f .,{ represents
equationsof .At-tre so-calledPenroseequations-furthertell us thatAt = At AAt andAAt = (AAl ). . Therefore.4t =
AIAAi - AAIAi - (AAlyAt = Al*A*At. Foreacha € C',hence0 ) Re(o',,{tc) = Re(o'r4f'l*Ala),thus
definitewheneverA is so. That the reverseimplicationis alsotme follows
completingthe proof of At being non-negative
tr
from the fact that (At;t - ,.
Pmblen 19-3: Characterizations Associatd with a Sum and h,oduct
hoposedby RonEnr E. HARTwrc, NorthCarolina State(Jniversity,Raleigh,NorthCarolinq IJSA,PETERSeIt{nL,
versityof MaribonMaribof Slovenit & HeNs JoecHtv WERNER,UniversittltBonn,Bonn,Gerrnny.
TheMitor hassofar receivedonlyone solution(in oddition to a solutianby thc Proposers)to thisproblzm-we lookforward
to receivingmoresolutions!
squarematricesA md B satisfyingAB = pA * gB, wherep andCaregivenscalars.
l. Characterize
linearoperatorsA andB actingon a vectorspace/ satisfyingABt e Span(.Ac,Bo) for
2. More generally,characterize
everya e N.
Pnoblem19{: Eigenvaluesof Positive SemidefrniteMatrices
University,Fort lattdcrdale,Florida, USAhoposedby FuzniN ZHANG,NovaSoutheastern
TheEditor hassofar receivedonlyone solution(from thePmposer)to thisproblem-welookforward to receivingmoresolutions!
x r matf,icesof
€ (0,1]-Oe € (0,1).u"nthat,if /s, ...,An arenon-negativedefiniter
ShowthatthereareconstantsT
rankonesatisfying
l . A r + . . ' * A " - . I "a n d
2. trace(A;)( 7 for eachi = 1,. . ., r,
of Ao = Dieo Ai allliein theintefval(e,1 - e)'
o of {1, 2,...,n} suchthattheeigenvalues
thenthereisa subset
Pmblem l9-5: Symmetrized h'oduct Definiteness?
proposedby txcuu OLKIN, stanford(Iniversity,stanford,califumio usA.
Let.4 andB be real synmeEicmatrices.Thenproveor disprove:
definite).
definite)then.48 * BA is positivedefinite(non-negative
L If A andB arebothpositivedefinite(non-negative
definite)'
definite)thenB is positivedefinite(non-negative
Z. lt A andAB* BA arebothposirivedefinite(non-negative
g9MMENT This problemis discussed
by peterLax on pp. l2o-121 of his new bookLircar Atgebra fwiley' 1997'ISBN
recentlyreviewedin Image,18,pp. 4-5. -Ed'
0471-11111-21
poge 29
April 1998
solution19-5.1by Davro GALLAN,universityof wisconsin,
Madison,wisconsin,
LJSA.
l. Thefirstassertionisfalse,
(l l) -o n = (i
evenforposiriveentrymauices.
Forexampte,r^tnr+,=
L/
l)./
6
\r
Then
\r
det(AB + BA) = -l andAB + BA fails to be non-negarive
definite.
2. The secondassertionis trueif / is positivedefinite.Sincex'(,,{,B + BA)x = 2x'ABx, the otberhypothesisimplies
x'ABx>0(>0)forallx#0.Iftheconclusionfails,thenBhasaneigenvalue.ls0(<0)witheigenvectorx,say.But
thenx'/Bx = .\x'.r{x < 0 (< 0), a contradiction.
O
solution 19-52 by sHuenczur Lrv, universitdtBasel,Basel,switzerland.
l. For positivedefinite(non-negative
definite)mahicesA andB , AB + B A is not positivedefinite(non_negative
defrnite)
in general.Herex'(AB + BA)x = 2a'ABs, for anyreal vectorc. An example:
n=( l,
\-z
-2 \
-_(
"-\0
2 ),
n o
t / .) ,
-10
\
ea+a,q=( 24
A B = ( , r , ;o, , /) ,
\_E
4 )
\_ro
- pyand;1(AB+BA)y
2
E,ybeitsassociatedeigenvector,
thenBy
=2{ABy - 2p{Ay.
,ry!*eaneigenvalueof
( I ) IF ,4 is positivedefinite,thenB is positivedefinite(ron-negative
Hence
definite)
whenever
,qA + A i ispositive
OJnnite
(non-negative
definite).(2)WhenA and,
AB * BA ue bothnon-negative
definite,B neednotbenon-negative
definite.An
example:
A- ( 3 00
B=(;
),
j,
)
AB*BA_(t
3)
Solution 19'5.3 by Davto LoNnoN, Tbchnion,Haifa, Israel.
l. Both parts of assertionI are false. Let A
\ 100 10
and B
300\ i- -", :^: .:,",
AB * BA
t7'-= ( ?0?
ls not non-negativedefinite.
IOO 404
\
\ 0
2 ).
A and B uepositive definite while
)
2 . Thepositivedefiniteversionof assertion
2 is tme. Withoutlossof generality,
wemayassume
thatB = d i a g ( D r , b z , , b n )
is diagonal.Let A = (o;i) ancl^/-B* BA = (ori$, + D1))be pisitive
definite.It followsthara;; > 0 and 2a;;b; > 0
f o ri = I , . . . , f l . H e n c e , 0) ; 0 ,i - 1 , . . . , n , i , i . ,B i s p o s i t i v e O e n n i t e .
3. The non-negarivedefiniteversionorassertion2isfalse.t*,r=
(3
(t 3)
l)
-.r=
is non-negative
definite.
(;t
l)
**
AB+BA=
o
solution 19'5.4 by HANS JOACHIM WERNER, Universitdt
Bonn, Bonn, Germany.
The following four observationsgive the complete solution
to this problem.
A andB areboth(reatsymmetric)
( I I andB - (; S) Ahhoush
)
non-negative
definite,thematrix AB + B A - l ' z 1 \ nus n benon-negative
definite.
observation 1: considerthe matric€sA =
\ i
O)
observation 2: considerthe matricesA =
(l l) andB- (; ,f;) .AhhoushAardBareboth(reatsymmetric)
positivedefinite,the mdilrixAB + I 7 A - / a
positivedefinite.
il ) faitsto be
[,,
poge 30
lmoge; No. 20
izConsiderthcnwtricesr=
observation
(l 3)"*"=
/,) n\
f ,
O)
(l
!r).o,*oushthenwtricesAandAB+BA-
*, both (real symmetric)twn-negativedefinite,the nwtrix B is not so.
Obscrvation 4z Izt A and B be real symmetrbmatrices.If A is positivedcfnite and AB * BA is rwn-negativedefnite
(positivedcfnite), then B is non-negativedefinite (positivedefnite).
pnoof of ObservatiDn 4: Recall that for any square(possiblycomplex) n x n matrix C its field of values(often also
- 1}. Sinceby assumptionAB + 8,4 is non-negative
calledits numericalrurge) is .F(C) '.- {a'Aa : o € An, o*t
definite(positivedefinite),the Properties1.2.5and L.2.6in RogerHorn & CharlesR. JohnsonlTopicsin Murix Analysis,
E RHP9 := {z € A : Re(z) 2 0} (respectively
CambridgeUniversiryPress,199U tell us that o(AB) g f@4
of AB, thatis'thepoint
o(AB)
denotesthespectrum
o(AB) g f(AB) q RHP := {z € C : Re(z) > 0});here
9'12 in R. B. Bapat
Exercise
(see,
for
example,
of the matrix .AB. From matrix theoryit is well known
sei or eigenvalues
fLircarAlgebraandLircarModels,HindustanBookAgency,Delhilgg3])thatifCandDuenxnmatricesthenCDand
DC havethesameeigenvalues.Hence,in particular,o(AB) = a(Ai BAi) whgre.Ai is tne uniquesquareroot of A. The
arereal. In view of o@\ ne*1 - o(AB) g REPo (respectively
matrix d,*n.l,* is symmeric andso all its eigenvalues
SinceAt ispositive
oeli Afi1= a(AB) g RHp)itnowfollowsttrat/i8,4* isnon-negativedefinite(positivedefinite)'
definite(positivedefinite).This
definite(rositivedefinite)if andonly if B is non-negative
oennite,,e,{Bl,*' isnon-negative
D
4.
completestheproof of Observation
solution 1g-55 by HENny WoLKowrcz, universityof waterlao,waterloo,Onnrio, catt^daLet A andB be real symmetricmatrices.
dennite),thenit is nottruethatAB * B-,{is positivedefinite(non-negative
L. lf AandB ue positivedefinite(non-negative
definite).
pmof. The proof follows (quickly) from using the following MAILAB pro$am (courtesyof Mike Overton,Courant
Instihrte,NY).
for
trY = 1:10,
X = diag(rand(10,1) );
Z = rand(lQ,10); Z = Z'*Zi
XZZX=y*g+Z*X;
t e i g ( x ) e i s ( z ) e i s ( x z Z X )l
Pause
end;
(1997) ['A note on the existenceof the AlizadehA specific example given by R. D. C. Monteiro urd Y. Zanjacomo
Programming'18:393401; trttp : / /www'
Haeberly-Overton direction for semidef,niteprogramming", Muhematical
is
isye, gatectr. edu/ l- nonteiro/tectr-reports/note'ps]
A-[ i o1s]andB=[;
,%]
D
of AB * BA are:-l and33'
of A ue: 8.1310and0.3690.Theeigenvalues
Theeigenvalues
thenit is not tme thatB is positivesemidennite:considerthediagonal
2. lt A andAB + B Aarebothpositivesemidefinite,
matrices
A=[l Sl,"dB=[3-0,]
B is positivedefinite'
But for A utdP - AB +B/ bottrpositivedefinite,thenit is tme that
of B. ThenA andut PU = (ut AU)D + !(vt!r/\re
proof. LetB = U DIlt be rheorthogonaldiagonalization
- AD + DA' But'
= D wasdiagonalandso w! Ir.tte P
both positivedeflnfte,i.e. we couldhave.rrut.o that B
tr
D to get D;; = P;;f (2Aii) ) 0, Vi'
thenwe cansolvethe consistentlinearequationin thediagonalmafix
April 1998
poge 3l
We addsomeobservations.
An immediatecorollaryto theproof of part 2 is
Corollery lz If A and AB + BA are nonaegativedefiniteandtlw diagorul of A is positive,thcn B is nonnegativedefuite.
In fact, thesetypesof questionsariseindirectly in certainalgmithmsfor solving semidefiniteprogramming(SDP)problems
suchas
min{tracecx : A,x- b, x > o},
whereX | 0 denotespositive semidefiniteness,C is a given symmetricmatrix urd is a given linear operator. These
",4
algorithms
stattwithtwogivenn x n symmetricpositivedefinitemaEices
Z,X > 0 andthescalarp= traceZ*1n.tney
thentakea Newton stepfor solving the nonlinearsystemof equations:
-0
('{34'
#)
)=:(
(1)
The first two equationsarelinearwhile the nonlinearityarisesfrom the third equation.This equationalsoresultsin the system being overdetermined,since it mapssymmenicmatricesto possibly tronsymmetricmatrices. One approachis to symmetrizethis equation,resultingin ZX + X Z - 2pI = 0. (Moregenaal symmetrizations
arediscussed
in R-D.C.Monteiro
andY. Zhang(1996)['A unifiedanalysisfor a classof path-followingprimal-dualinterior-pointalgoritlrmsfor semidefinite
programming",Tbchnicalreport"GeorgiaTbch,Atlanta,GA, 1996,ftp: / /pc5 .math. umbc. edu/pub/mz-sdp. ps . gz or
fEpt / /pc5.math.r.rnbc.edu/pub/mz-sdp.dvi.gzl.) Thelinearization
of theresultingsquaresystemhasa uniquesolution (calledtbe AHO direction;seeF. Atizadeh,J.-P.A. Haeberly,andM. L. Overton(1996)
[,.primal-dualinterior-point
methodsfor semidefiniteprogramming: convergencerates,stability and numericalresul6", NyU ComputerScienceDept
TbchnicalReport721,CourantInstituteof MathematicalSciences,
to appearin SIOpTI) it Z X + X Z ispositivesemidefinite. (This is shownin M. Todd,K. C. Toh, anclR. H. Thtuncu(1996)
directionin semidef,nite
["On the Nesterov-Todd
programming",TbchnicalReportTRll54, Schoolof oR andIE, CornellUniversity,Ithaca,
Nyl andM. Shida,S. Shindoh
andM. Kojima ( 1996)["Existenceof searchdirectionsin interior-pointalgorithmsfor the
SDpandmonotoneSDLCF,,Tbchnicalreport"Dept. of InformationSciences,
Tbkyo Instituteof rechnology,Tokyo,Japanl.)However,astheaboveexample
shows,this may not hold.
However,thefollowingisclear:it ZX - pI = 0(i.e. Z,X lieonthe so-calledcenralpath),
tbenZX + XZ > 0, since
commuhtivitymeansconunonorthogonaldiagonalization.
Therefone,
a naturalquestionto askis:
Pmblen lz GivenZ,X > 0, p)
0,defineaneighbourhoodof
thecentralpath
llzx - plll < eor llZiXz* - p41< e
whichguaranteesttnt ZX + XZ > 0.
Sucha resultis discussed
by Monteiro& zanjacomo(1997)op. cit.,who showthat
llz*xz* - prll< p/2
impliesthatthelinearizationhasauniquesolution.Infact,thisdoesnotimplyXZ+ZX
>0asOeaboveexampleillustrates
with v - 8' However,thetriangleinequality(observation
dueto Mike To<lct)
<toesshowthat:
llzX - plll <p impliesZX + XZ > 0,
thusprovidinga neighbourhood
ofthe centralpath.
o
solutionswereako receivedfrom Rocpn A. HoRN, llniversityof (Itah,
saltIalrc city, utalr, usA; cnr-KwoNc Lr &
RoY MATHIAs' Thecollegeof williamandMary, witliamsburg,vrgini;
usA; Jonue KAARLgMERrKosKr& ARr vrRTANEN,universiryof ranpere, Tampere,
Finland;FERNANDoc. irlve, IJniversityof Lisbon,Lisbon,portugat;DAFNA
Sttesue, Tbl'HaiRodmancoltege,IlpperGatitee,IsraeliYoNccE TtaN,
concordiaIlniversity,Moftreat, eucbec,canada;
andfrom theproposerIN.RAM OLKTN,StanfordUniversiry,
StunfordCA, USA.
I
lmoge; No. 20
poge 32
New Problems
Problen 20-1: E[envalues
hoposedby BeNrToHnnxANnrz-BERMEro, UniversidadNacioml de Hucacifin a Disftncia Madrid, Spain.
l-etR,pandKbenxnrealmatrices,wherePisdiagonalandinvertible,Kisskew-syrnmetricandft=PK.Provethat:
l. The eigenvaluesof R appearin pairsof oppositesign of the form *lf, where{ is areal numberwhich maybepositive'
zero or negative.
2. If the diagonalentriesof P areall positive or all negative,thenthe eigenvaluesof .Rappeu in pairs of oppositesign of
the form *rf€, wnere{ is nonpositive,i.e., all eigenvaluesarezero or imaginary.
Problcm 20-2: Eigenvaluesand eigenvectorsof a patterned matrix
Proposedby EuIl-to SPEDIcATo,Universityof Bergmw,Berganw,Italy'
= 1,2,"',n)'{Forthe
A= {o;il = {li-il]-.(i'i
x npatternedmatrix
Findtheeigenvaluesandeigenvectorsofthen
(
i
,
i
=
!
,
2
,
.
.
.
,
n
)
,
s
e
e
P
r
o
b
l
e
m
l
S
5
i
n
I
mage,19'28-30.1
e i g e n v a l u e s a n c t e i g e n v eAc =
t o{ ios;oi }f= { ; - i 1 '
Problem20-3: When is {XlX' ( ,'{} Convex?
USA.
hoposedby G. E. tnepp & BoHE WANG,WestVirginia(Iniversity,Morgantown,WestVirginiL
matrices:
dennitematrixA andpositiveintegqn,let C(n, A) be oe followingsetof Hermitian
For aHermitiannon-negarive
C(n,A) -- {XlX = X*,
X" S A}'
Forwhatvaluesofn=l,2,...isthesetC(n,A)convex?HereSdenotestheL0wnermarixpartialordering'
Problen 20{: Continuity of T[ansposition?
proposedby Gorz TRexxLeR, universittltD\rtmwtd, Dortmund,Germany.
thet)'peof A suchthat '4" = DLt Di=' B'.iAB;i '
l,et A be a real rn x n matrix. Find mauicesB;i dependingonly on
multiplicationof matrices'it would
of .4. [If suchmatricesexisted,by thecontinuityof additionand
where,4" is thetranspose
provealsothecontinuityof transposing'l
Problem 20'5: Matrix SimilaritY
University,Fort laudcrdole' Florida' USA'
proposedby FUZHENZuanc, N:ovaSoutheastern
thentheyaresimilar overthereal numbers'Show
numb€trs,
As is well known,if two real matricesaresimilar overthecomplex
ue unitaritysimilu overthe complexnumbers'
hordsfor unitarysimiluity. *ecisery, if two real matrices
that this assertion
realnumbers'
thenshowthattheyareorthogonallysimilu overthe
SquareRoots
Problem 20'6: Non'negative Definitcnessand
iuriurrsity, FortI'auderdale,Florida, usA'
southeort
Nova
proposedby FuzHENf"^*o,
I.,etA>0, B > C Z 0. Proveor disProve
BLln
z g t l z > C r l zA c r l z ;
here> is theLownermatrixpartialordering.
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west'Montrear'
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ooot,rueShsbrooke
universitn'dos
T ff.""li"l11Tsiltisiicr,'r,r"cli
to Problems18-1,19.3and 194!
We look fonr,ardparticululy to receivingsolutions
2K6.
Qu6bec,CanadaH3A
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