ETNA
Electronic Transactions on Numerical Analysis.
Volume 36, pp. 99-112, 2010.
Copyright  2010, Kent State University.
ISSN 1068-9613.
Kent State University
http://etna.math.kent.edu
THE STRUCTURED DISTANCE TO NEARLY NORMAL MATRICES
LAURA SMITHIES
Dedicated to Richard S. Varga on the occasion of his 80th birthday
Abstract. In this note we examine the algebraic variety of complex tridiagonal matrices , such that
, where is a fixed real diagonal matrix. If then is , the set of tridiagonal normal
, we identify the structure of the matrices in and analyze the suitability for eigenvalue
matrices. For estimation using normal matrices for elements of . We also compute the Frobenius norm of elements of ,
describe the algebraic subvariety consisting of elements of with minimal Frobenius norm, and calculate the
distance from a given complex tridiagonal matrix to .
Key words. nearness to normality, tridiagonal matrix, Kreı̌n spaces, eigenvalue estimation, Geršgorin type sets
AMS subject classifications. 65F30, 65F35, 15A57, 15A18, 47A25
1. Introduction. In this note, we establish a generalization of the matrix nearness problem which was solved by S. Noschese, L. Pasquini, and L. Reichel in [5]. The structure for
the type of generalization of nearness to normality which is considered in this note was first
suggested to me by Roger Horn, in connection with [2]. The homework set in [3, page 128]
also discusses this type of generalization of normality.
Let denote the set of all real !#"! irreducible tridiagonal matrices and let $% denote
the algebraic variety of real normal irreducible tridiagonal !&"'! matrices. The paper [5]
presents the following, for any fixed (*)+ :
(i)
a formula for Frobenius distance ,.-/0(12$354 and an easily calculated upper
bound on this distance;
6 7 - is
(ii) a formula for a real normal tridiagonal matrix ( 6 , such that 78(:9;(<
equal to , - /0(12$=54 ;
(iii) simplified versions of (i) and (ii) for Toeplitz matrices.
In order to generalize the above problem, we must fix some notation. Let >=?A@B? denote
HGJI
the set of complex !C"! matrices and let DE)F>3?A@? . Define the adjoint of D , D
G D*K ,
D
D
to be the conjugate transpose of D . Recall that D is defined
to
be
self
adjoint
if
ONQPRG
and normal if the commutant of D and its adjoint, L D*1MD
DSD
9TD D , equals U in
>?A@? . Throughout this note, fix a real diagonal matrix:
(1.1)
V
GXWHY[Z]\
G
/2^_`4a1 such that VSb c
U 1^edfg^AhifXj5jkjBfg^ ? 1
and
lm ?
n d
^
m
Gporq
Let denote the set of tridiagonal complex matrices. The purpose of this paper is to
investigate the set
$es
GSt
Du)v
P
DSD
9TD
D
G
Vwx1
and to describe the distance from normality for the elements of $cs . More precisely,
Received March 13, 2009. Accepted for publication August 9, 2009. Published online on January 20, 2010.
Recommended
by L. Reichel.
Department of Mathematical Sciences, Kent State University, Kent, OH 44242
(smithies@math.kent.edu).
99
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100
LAURA SMITHIES
we provide simple formulas, in terms of V , for the elements of $ s ;
h
for (*) , we give a formula for the Frobenius distance , - /y(1z$es4 , and an
easily calculated upper bound for this distance;
(iii) we provide formulas for the elements of the subvariety {;s of $es whose
Frobenius norm is minimal, and for the unique element D|s}){~s with
only nonnegative entries;
(iv) we combine the above results with those of [2] to describe for any DE)+$s
the distance to normality both in the Frobenius norm and in the sense
of the suitability of D for eigenvalue estimation through normal matrices.
This paper is organized as follows. Section 2 recalls some elementary results and introduces notation which will be used throughout the paper; Section 3 presents a characterization
h
of the elements of $s . In Section 4, we give a formula for the distance , - /0(12$es4 from
(€)| to $es , and we describe the algebraic variety {;s of the elements of $s of minimal
Frobenius norm. In Section 5, we describe the distance from normality for the elements of
$ s , in part, by applying results from [2]. The final section discusses some conclusions and
possible extensions.
(i)
(ii)
2. Background and notation. This section defines notation used in the sequel and recalls some elementary results. Table 2.1 collects our most important notation.
TABLE 2.1
Sets used in this paper.
G
the tridiagonal matrices in > ?@?
‚
G
† G
ˆ
self adjoint ( ‡
G
G
† ƒ G † „
‡ ) in >?@?
G
anti-self adjoint ( ‡
G
$
‚ƒ G ‚…„
the normal matrices in >?A@B?
ˆ ƒ G ˆ „
9‡ ) in >?A@?
‚vƒ
real, irreducible matrices in
GgWY‰ZŠ\
V
o
/z^ _ 4 , ^ d f:^ h fj5jkjBf&^ ? , ^ dŒ‹
GSt
P
8NeG
$es
DŽ)
L D*1MD
Vw
$e s
‘
s
Gt
G*t
Du)v
/0‡1M’“4”)
P
8NeGg
L D*1MD
† ƒ}• ˆ ƒ
P
NG
G—™˜Y
¡
¡
(
¡ ›`d
¡
¡
¡ šed
¢
?n d ^
Ggo
.
9H–]VHw
Let (
/0še1M›œ1žr4 denote the element ( of > ?.Ÿ d and diagonal ›<)> ? . That is,
(2.1)
, 
Vw
L ‡1’
G
m
m
0
`d
›5h
..
.
..
.
..
.
š ?.Ÿ d
with lower and upper bands š and  in
0
£¥¤
¤
¤
¤
¤
¤ q
 ?.Ÿ d ¦
› ?
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THE STRUCTURED DISTANCE TO NEARLY NORMAL MATRICES.
G
9‡
Note that we use the terminology anti-self adjoint to refer to the constraint ‡
G
9‡ , and the terminology
on ‡…)>?A@B? . When ‡…)*§?A@? this simplifies to ‡HK
antiˆ
symmetric and skew-symmetric are also used. We use a superscript, as in
, when our
matrices are constrained to be real.
‘ We remark that, in contrast to the sets $ defined in [5], the matrices in our sets $ s
and s are allowed to be reducible. However, the choice to arrange the entries of V in nonincreasing order, plays the role of irreducibility. More precisely, cf., [8, page 11], let ¨ be
tª©
a permutation of 1kjkj5je1ž!w and let « denote the corresponding permutation matrix, i.e.,
Gp®
«_M¬ ­
_M¬ ¯ª°‰­8± . We say that a matrix ‡)v>?@? is reducible if there exists a rearrangement of
coordinates with respect to which ‡ is block diagonal. That is, if there exists a permutation
«²)v§?A@? , such that
G´³ Œ
q
‡ o d8¬ dµ‡Œd8¬ h
‡Hhk¬ h#¶
WY‰ZŠ\
}G·WY‰ZŠ\
ONG
It is easy to check that «
/z^ ¯ª°R_± 4 and that L D*1MD
V if and
NGWHY[Z]\ /z^_`4ž«
only if L «D« 15/2«D« 4
2
/
^
4
. Our conditions on V imply that V cannot equal
ª
¯
R
°
_
±
WHY[Z]\
‘
/2^ ¯ª°R_± 4 for all permutations ¨ , and consequently, we do not limit our solution sets s
and $ s to irreducible matrices.
Let ‡1’¸)v>?A@B? . Recall that the Frobenius inner product and induced norm are defined
«i‡H«
as
For »
G
Gp—™˜
GSº
q
/0‡<1’+4¹/2’ ‡ 4a1 and 7a‡“7k/0‡1M‡4žG
/0»_`4ž_? n d and ¼
/y¼`_½4ž_? n d in >? , their Euclidean inner product is
G l ?
q
» _ ¼`I _
_n d
GÀ¿ ¾
The corresponding vector norm is 78»75h
»c1ž» ‹ and the induced operator norm is
GpÁÂBÃtÄÅÄ ÄÅÄ P
GS©
7a‡+7ah
‡» h
7a»Q7kh
w.
ˆ
†
Let ÆÇ and ÆÈ ¿ denote the projections of >?A@? onto and , respectively. That is, for
G
©
DŽ)v>?A@? and É
9 , let
¾
Æ È /2DS4
G
DËÊÌD
–
GXÍÏÎ
»™1ž¼ ‹
/zDS4
and Æ Ç /2DS4
G
DÐ9TD
–
G
ÉÒÑÔÓ/zDS4
be its self adjoint and anti-self adjoint parts. It is easy to check that
> ?@?
G † • ˆ
1
•
† „ ˆ
G
HG
Õ and Õ
9HÕ .
where G*denotes
the direct sum. More precisely, if ÕÖ)
then Õ
o
So, Õ
. This, combined with the fact that both the set of self adjoint matrices and the set
of anti-self adjoint matrices are closed under subtraction, implies that the decomposition of
)&>?A@? into
any D
ˆ ƒ of a self adjoint and an anti-self adjoint matrix is unique. The
† ƒ<a • sum
natural structure on
, from the point-of-view of functional analysis, is that of a Kreı̌n
space, as we discuss
in
Section
5.
ˆ
†
The sets
and
are
orthogonal
with respect to the Frobenius inner product. However,
ˆ
†
† • ˆ
GWHY[Z]\
this is not true of and . For example, if ×
/0›a_½4)}† §?A@B? , thenˆ /2×1žÉÔ×Ø4H)
Gٗc˜
G
and /0×1ÉÒ×4¹and ’·)
then /2‡1’+4ž- is
/¹9ÏÉÔ× h 4
9ÏÉ87k›B7 hh . In G general, if ‡Ú)
—™˜
G —c˜
G
a pure imaginary number. Indeed, /2‡1M’“4because /2’ ‡4
9
0
/
‡
M
1
“
’
ž
4
/0‡ ’+4
—c˜
G
—™˜
q
/¹9‡’ 4
9
/0’ ‡Œ4 On the other hand, the properties of the trace function imply that
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LAURA SMITHIES
h G
ˆ 7† 7a‡|9’
and
ÍÏÎ
q
7a‡“7 h- 9–
//0‡<1’“4 - 4eÊÌ7k’7 h- Thus, sums and differences of matrices from
also satisfy the Pythagorean constraints:
h G
7a‡|Û|’7 -
(2.2)
† • ˆ q
h
h
7k‡“7 - Ê 7a’7 - 1Ü/0‡<1’“4%)
Notice that D is tridiagonal if and only if D
is, and this happens if and only if the
self adjoint and anti-self adjoint parts of D are tridiagonal. Moreover, a routine calculation
shows that for any DŽ)>?@? ,
L D*1MD
(2.3)
NeG
9H–BL ÆÈ/2DS4a1ÆÇ/zDS4
NÒq
Finally, we measure the (squared Frobenius) distance from a matrix D
subset Ý of >?A@? as
)*>=?@? to a
GgY‰Þß5t
GpY‰Þßkt
q
h P
h P
h
, - 2/ D*1MÝØ4
78àØ7 - DáÊ|àâ)vÝw
7aDÐ9CàØ7 - àâ)Ýw
† • ˆ
† • ˆ
Equivalently, for any /0‡1M’“4”)
and ÝÚã
define
g
G
‰
Y
Þ
5
ß
t
q
h
h P
h
(2.4)
, - //0‡<1’“4a1MÝ4
78‡&9CàØ7 - Ê78’S9Cä|7 - /0à1ä}4”)Ýw
G † • ˆ
Note that equation (2.2) implies that the equivalence of >?@?
which is given by
DæåE/0ÆHnj/2DS4a1ÆHÈ3/2DS4ž4 and /0‡1M’“4åЇpÊ&’ is an isometry with the natural choices of
topology.
3. Structure of $ s . Recall that we have fixed the non-zero, real diagonal matrix V
with entries which satisfy the conditions (1.1). In this section we will study the structure
ONG
of tridiagonal complex matrices D
)S , such that L D*1MD
V . We shall see that the
computations involved simplify with the following equivalences:
‘
L EMMA 3.1. Define the sets $ s , $e s , and s as in Table 2.1. Then
()ç$ s
if and only if –œ(*)ç$e s 1
and
DE)ç$e s
if and only if /2ÆHÈ3/2DS4a1ÆHnj/2DS44%)
‘
s
q
Moreover, for every äè)>?A@? ,
 h
G h
G h
‘
q
, - /0äF12$ s 4
, - /z–]ä1z$e s 4
, - //0ÆHÈ3/2–]ä#481ÆÇ/z–]ä}4481 s 4
Proof. The map (²åé–]( obviously defines a bijection between $ s and $e s . Equation
‘
(2.3) yields the equivalence between $ s and s . Now, let ä…)v> ?A@B? . Then,
G
Y‰Þß
h
h G&²YÅÞß
h G& h
, - 2/ –]äF12$e s 4
, - 0/ äF12$ s 481
hMêë`ìœíÒî 7k–]äï9T–œ(“7 êë`ì]î 7aäÚ9'(“7 and the Pythagorean property (2.2), implies that
G h
‘
q
h
, - 2/ –Šä12$e s 4
, - /ž/2ÆÈ3/z–]ä}4a1ÆHnj/2–Šä}4ž4a1 s 4
GXWHY[Z]\
G©
D EFINITION 3.1. Let V
/2B
^ _½4 be as in (1.1). For each ð
1kjkj5j1ž! , define
m
_
ñ G lm
q
^
_
n d
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THE STRUCTURED DISTANCE TO NEARLY NORMAL MATRICES.
8N
)>?A@B? , L D*1MD MN3G is self adjoint and has
R EMARK 3.1. We remark that for any D
trace zero. Thus, our fixed right hand side in the equation L D*1MD
V is, up to choice of
ordering for the real numbers ^B_ , the general non-zero, diagonal right hand side.
The motivations for choosing a diagonal matrix for the right hand side V will be discussed
in the final section. The essential point is that we are interested in describing the extent to
MN
which a matrix D fails to be normal in terms of the rank of the commutant L D*1MD
. Our
motivation for assuming that the ^B_ m are in non-increasing order is indicated by the following
lemma.
ñ
GXWY‰ZŠ\
L EMMA 3.2. Let V
/2^ 4 satisfy condition (1.1) and let _ denote the ð -th partial
sum. Then
ñ
o
_i‹
1 for all ð
m
G
G©
1kj5jkj1ž!v9
©Šq
m
o
ñ
¾
o
Vïb U , we must have ^ d؋ . Now let ð be minimal,
Proof. Since
such that _
.
G
¾ mo
¾ o
^
^
ò
ª
ð
k
1
5
j
k
j

j
ž
1
!
Then ^ _ ñ . Since
the
are
non-increasing,
this
means
for
all
.
ñ GXo
¾ ñ ¾ o
Therefore,
_
for all ò ‹ ð . However, by assumption
.
† ƒ
?
We
can
now
simplify
our
study
of
the
structure
of
the
matrices
in
$ s . Let
(respecˆ ƒ
tively
) denote the self adjoint (respectively anti-self adjoint) matrices in , the complex
tridiagonal matrices. Recall that we have defined
‘
s
G*t
† ƒ}• ˆ ƒ
/2‡1M’“4”)
P
L ‡<1’
NG
9H–ŠVw
q
‘
Gôóõö
s corresponds to a unique (
Because of Lemma 3.1, each /2‡1M’“4Œm )
h in $ s . Thus,
s
the following lemma yieldsG€aWcharacterization
of
the
elements
of
.
$
† ƒ • ˆ ƒ
Y[Z]\
/2^ 4 satisfy condition (1.1). Let /0‡<1’“4)
L EMMA 3.3. Let V
and
denote the entries by
‡
Then /0‡<1’+4”)
(3.1)
‘
GX—c˜MY ÷ I
/ _ 1ø _ 1 ÷ _ 4 and ’
GX—c˜MY
s if and only if the entries satisfy
ÍÏÎ ÷ ù I
G ñ
/0ɹ4
/ _ _`4 G _œ1
÷
ù
/0ÉÒÉÔ4
_ _ õ d G ù _ ÷ _ õ d]1
/0ÉÒÉzɹ4
øŠ_ G øŠ_ õ dœ1
/0ÉÒü.4
ú_
úM_ õ d`1
q
/¹9 ù I _ 1žÉ¹ú _ 1 ù _ 4
GS©
ÜBð S
G ©
Ü ð S
B
G ©
Ü ð S
B
G ©
Ü ð
B
©Šû
1kj5jkj1!9 û
1kj5jkj1!9T– ©Šû
1kj5jkj1!9 ©Šq
1kj5jkj1!9
G
G
9 ’ , the diagonal
Proof. Note that since ‡
G ‡ , its
N diagonal /0ø _ 4 is real and since ’
Õ
L
‡
M
1
’
‡
’
Õ
of ’ is imaginary.
Let
.
Since
and
are
tridiagonal,
is pentadiagonal.
G
Moreover, Õ
Õ . A routine calculation shows
Õ_M¬ _
Õ_M¬ _ õ d
Õ_M¬ _ õ h
G
ÍÏÎ
ÍÏÎ ÷
N
9H–BL / ÷ _ ù I `_ 4Q9
/ _ Ÿ d ù I _ Ÿ d½4 1
/0øŠ_9'øŠ_ õ d54 ù %
_ ÊýÉO/zú_ õ %
d 9Tú_½4 ÷ _]1
÷ _ ù _ õ d”9 ù _ ÷ _ õ d q
G
G
Equating Õ to 9H–]V , it is easy to see that conditionsñ (i) and (ii) of (3.1) hold. Recall that V
ÍÏÎ ÷ ù I
G
G²o
G¸©
©
is assumed to be non-zero and hence
/ _ _`4
_b , for all ð
1kj5jkj1ž!}9 . Thus,
o
Ggo
÷ _+Gp
ù
and _þb
for all ð . The remaining equations say
b
/0øŠ_ õ dÏ9'øŠ_½4
G
÷_ G
ÉO/zú_ õ d%9Tú_54 ù
ÉO/zú_ õ d%9Tú_54
_
÷_ù _ q
Äù Äh
_
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LAURA SMITHIES
The left hand side is real, the right hand side has non-zero imaginary part, unless ú _ is constant. Therefore, ø _ and ú _ are† constant.
ƒ• ˆ ƒ
G
N
Conversely, let /0‡1M’“4”)
satisfy conditions (3.1), and let Õ
L ‡1o ’ . Clearly,
G
X
G
o
G
p
G
÷ _ ù _ õ dB9 ù _ ÷ _ õ d
Õ3_¬ _ õ d
/2ø]_B9iøŠ_ õ da4 ù _.ÊØ/zú_ õ dx9ú_54 ÷ _
and Õ3_M¬ _ õ h
. Moreover,
Õ _M¬ _
‘
G
ñ
ñ
NG
9H–BL _ 9 _ Ÿ d
ÍÏÎ ÷ ù I
ÍÏÎ ÷
NG
/ _ _ 49
/ _ Ÿ d ùI _ Ÿ d 4
9H–BL
q
9H–ª^ _
s .
Thus, /2‡1’+4%)
‘
In view of the previous lemma, we denote the entries of an element /0‡<1’“4 of s by
G
®
®
Gg—™˜ Y ÷ I o ÷
Gg—™˜Y ù I o ù
/0‡<1’+4
/ÒÿvÊø 1èÊvɹú 4 , where ø1Mú)v§ , ÿ
/ _ 1 1 _ 4 and
/¹9 _ 1 1 _ 4 .
Note that
NG
L ‡1M’
LRÿýÊÌø
®
1ËÊýÉÔú
®]NG
‘
NÒq
L ÿ1
G
G
o
L 1M–c4
With this in mind, we can now describe the elements of s for V²b U . Let õ
denote the torus, and let §
denote them positive real numbers. We will use the both of the
elements
notations and tom identifyG´
ñ
WY[Z]\ of .
/2^ 4 satisfy
T HEOREM 3.4. Let V
(1.1), and let _ to be the ð -th
• ˆ ƒ
† ƒ condition
‘
partial sum of the ^ . The ordered pairs from
which lie in sÄ are
parametrized
õ
ù d Ä 1
Šd`1kj5jkj1
dk4
bijectively by §ç"ا
"=?.Ÿ d . Specifically,G each /y!+® Êx4 -tuple
0
/

ø
M
1
œ
ú
œ1
1
?.Ÿ
G
®
defines the complex, tridiagonal matrices ‡
ÿýÊ|ø and ’
Êýɹú , where the entries
of ÿ and
satisfy
(3.2)
and
GÖ©
ù _ G Äù _ Ä
÷ _ G#" ù _ 1
/yɹ4
/yÉÒɹ4
Ä Ä h G
! 1
where ù _ !
"G
õ
$
1
where
1
©
15jkj5j1ž!9 . Ä Ä
for all ð
õ
"&3?.Ÿ d , and
Proof.® First, let /0® ø1Mú`1œ1 ù d 1%
d 15jkjkj1
?.Ÿ d 4 , be a fixed element in §i"ç§
let /ÒÿÌÊÌø 1ÐÊ|ÉÔú 4 be defined by plugging this /y!Ê'ª4 -tuple into the given formulas. It
G
is easy to check that ÷ d ù I d
^ed=ÊÌÉ , and that conditions (i)-(iv) of Lemma 3.3 hold. Thus,
‘
/0‡<1’+4”) s .
‘
G
®
®
s ©. By the previous
Now, let /2‡1M’“4)
/2‡ñ1’+4
/zñ ÿÌÊ:
ø o 1ÐÊýɹú 4 for some
G
€
G
Gô
© ÍÏÎ lemma
ù I _½4
all ð
_ and _Cb , by Lemma 3.2. So,
ø1Mú)&o § , and for G·
1kj5jkj1ž© !#9 , / ÷ _ G)
©
(H˜\ ù
ù _ýGÚ
÷ ùI
for all ð
b
15jkj5j1ž!'9 . Define 5_
/
_`4 and let
©
Ä ù Ä denote ÑÔÓ/ d d54 . We
will show that /zÿ1S4 is given by plugging the /0!çÊ 4 -tuple /*œ1 d 1
]dœ1kj5jkj1%
?.Ÿ dk4 into the
‘
s ,
above formulas. Since /ÒÿÏ1S4”)
÷ _ G
ÍÏÎ ÷ ù I G ñ
/ _ ½_ 4
_Š1
Now,
ÍÏÎ
÷ _ G
/ù 4
_
ÍÏÎ ÷ ù I
/ _ ½_ 4 G
Äù Äh
_
Äù
ñ
_
_
Ä h 1 and
and ù
_
Äù
_
Äh G
ÍÏÎ ÷ ù I
/ d 5d 4 G
Äù Äh
d
÷ _ GgÍÏÎ ÷ d G
/ ù 4
/ù 4
_
d
ÍÎ
Thus,
÷ _õ d q
ù _õ d
ñ
Ä Ä
_ ù d h q
^ed
Similarly,
÷ _ G
÷ d G
ÑÔÓv/ ù 4
ÑÔÓF/ ù 4
_
d
ÑÔÓF/ ÷ d ù I da4 G
Äù Äh
d
Äù
d
Äh
q
Äù
^d q
Ä
d h
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THE STRUCTURED DISTANCE TO NEARLY NORMAL MATRICES.
ÍÏÎ
Combined with our second formula for
/,+ 4 , this tells us that
÷_ G
GÖ©
©Šq
^ed3ÊýÉ ù
Äù Äh
15jkjkje1!9
_ for all ð
d
Ä Ä
Ä Ä
õ
d
d
Finally, suppose /0ø1Múœ1œ1 ù d 15*/ k_œ4 _ ?.n Ÿ d 4 and / ø
1 ú½6 1 œ6 1 ù 6 d 15/ k6 _½4 _?.n Ÿ d 4 in §-"F§
".3?.Ÿ d
6
† ƒ • ˆ ƒ
G
G
G ÷
6 4 in
ø6 , ú
ú6 , ÷ _
define /0‡<1’+4 and /3‡<6 1%’“
which are equal. Then, ø
6_
G
G
©
Š
©
q
/
G
H
(
˜
\
/
G
(
M
˜
\
G
ù _ for all ð
ù _`4
ù _`4
6
and ù _
Thus,
,
and
5
1
k
j
5
j

j
1
|
!
9
½
_
/
/
5
_
6 G
6
m
G
G q
ÑÔÓ/ ÷ d ù dk4
ÑÔÓF/ ÷ 6 d ù 6 d54 G*
6
WHY[Z]\
/2^ 4 satisfy condition (1.1). Each Dµ)Ø$s is uniquely
C OROLLARY 3.5. Let V
Ê ª4 -tuple
determined by an /0!0
õ
Ä Ä
d q
/2ø1Mú`1 œ1 ù d %1 Šd`15jkjkj1 ?.Ÿ d54%)§ "§
" ?.Ÿ
1
Specifically, each such /y!ØÊ0ª4 -tuple defines the complex, tridiagonal matrix of the form
D
"G
!
õ
©
G
–
¿
G
—™˜Y
"
©
"
©
/ž/ I 9 4 ”
2 1MøŒÊýɹú`15/ Ê
42 481
d
7 ! 4536 4%)v> ?.Ÿ .
3$ , and 2
where
/
‘
õ
s and
Proof. By the previous theorem, we know how § "F§
"8 ?.Ÿ d parametrizes
G óõcö
s
is uniquely defined by D
for
some
we know from Lemma 3.1 that each D
#
)
$
h
‘
/0‡<1’“4Ï) s .
4. Distance formulas. In this section we establish a formula for the elements D s in $es
of minimal Frobenius norm. Specifically, we show the minimal elements form an algebraic
d
subvariety { s which is isomorphic to =?.Ÿ . We also define D s the unique matrix in { s
with nonnegative entries. Equivalently, 9HD s is the unique ÿ -matrix, cf., [8], in { s . We
also give a formula for the distance from a fixed ()v to $ s , and an easily computed upper
bound on this distance.
First we need some preliminary calculations on the behavior of the Frobenius norm on
tridiagonal matrices. Let ()v and write
(
Gp—™˜MY
/0še1M›œ1žr4a1F×
GpWY‰ZŠ\
/2› _ 481 and D
GX—c˜MY
o
q
/2še1 ž1 r4
Then
h G
78(“7 -
(4.1)
h
h G
7a×#7 - Ê 7kD¸7 -
h
h
h q
7a›B7 h X
Ê 7kš37 h Ê78c7 h
In particular,
h
h q
7k›B7 h |
Ê –A7aš37 h
—c˜
G
D 4
Formula (4.1) follows from the straightforward calculations that /0× S
7a›B7 hh , and
if ‡
(4.2)
—c˜
/2D
DS4
Gp—c˜MY
I 1 ›œ1Mš481 then 7k‡“7 h- G
/ÒÛ š
o
G
h
, 7k×}7 -
.? l Ÿ d
G l ?
G l ?
q
D M_ ¬ _ d D'_ Ÿ d8¬ _ ÊÌD _M¬ _ õ d D#_ õ dO¬ _
. _ Ÿ d88_ Ÿ d3Ê
š_5š._
Ÿ
_n d
_n h
_n d
G
Now we are ready to find the minimal elements of $ s . Let D )'$ s and let /0‡<1’+4
‘
Gp™
— ˜Y ÷ I
/0ÆHÈ3/2–ŠDS4a1ÆÇH/2–ªDS4ž4 denote the corresponding element in s . Write ‡
/ _Š1MøA1 ÷ _½4 and
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GX—c˜MY
’
LAURA SMITHIES
/ž9 ù I _ 1žÉ¹ú`1 ù _ 4 , where the entries are defined as in condition (3.2). Then by equations
(2.2) and (4.2),
h G
75D¸7 -
(4.3)
h
h G h
h
h
h q
7k‡“7 - Ê 7a’7 ø !Ê|–A79A7 h ÊÌú !Ê|–A727 h
Clearly the norm of D is minimized by the !choice of zero for the diagonals of ‡ and ’ .
õ ù
G
Äù Äh G Äù Äh
! 3$ _ and _
d . Define
Moreover, the relations (3.2) imply that ÷ _
m
l Ÿ d ñ G .? l Ÿ d l m _
:|G ?.
q
_
^
_n d
_n d n d
h
To minimize 7kD¸7 - , we need to minimize
(4.4)
?.l Ÿ d ^ hd Ê0 h ñ _ Ä ù Ä h
Ä Äh G
×<;x/ ù d 4
–BL
/ Äù ď 4
d Ê
^ d
d
_n d
Ä Ä G
This is minimized at ù d h
ñ
_ Ä ù Ä h NG
d
^ d
–
Ä Äh q
^ hd 0
Ê h
/ Äù Äh Ê ù d 4
^ d
d
:
º
^ hd Ê= h , and the minimal value is
 : º h
?
G
^ d =
Ê h q
×;x%/ > ^ hd =
Ê h 4
^ d
let
) º$ s be defined by evaluating the formulas in Corollary 3.5 at
Ä D Ä h G:
GXTo
o summarize,
GXo
^ hd Ê= h , and /@
d 15jkjkj1
?.Ÿ d 4%)A=?.Ÿ d . Then
,ú
, Œ)v§ , ù d
: º
^ hd =
Ê h q
h G
75D¸7 ^ d
G²o
GÖo
Ä Äh G
^ d imply
This is, of course, minimized
when . The choice and, hence, ù d
! õ
"'G
©
3$ equals . Combined with Corollary 3.5, the above observations
that the factor
m
m
m
establish the following theorem.
ñ G
GXWHY[Z]\
_
 n d ^ and
T HEOREM 4.1. Let V
/2^ 4 satisfy condition (1.1), and define _
:G
:
d ñ
 _?.n Ÿ d _ . The minimal Frobenius norm of the elements of $cs is . The subvariety of
:
>?A@? consisting of elements of $ s which have norm is
P
G*t
GX—c˜MY o o
º ñ
d q
{~s
D s
/ 1 1 @/ d 1kjkj5j%1 ) ?.Ÿ w
_ 4
?.Ÿ d 4%.
ø
In particular,
DCs
Gp—c˜MY o o º ñ
/ 1 1
_ 4
B
G
:
h
is the unique element of $ s with 7kD s 7 , and nonnegative entries.
Recall that we measure the (squared Frobenius) distance from a matrix D
of >?A@? by
GgY‰Þßkt
h P
h
, - z/ D*1MÝ4
7aDé9'àØ7 - à
)Ýw
to a subset Ý
q
Given any fixed (ï)Ì , we want to find the distance from ( to $cG s . Equivalently, (up to a
factor
2/ ƌÈ/2–œ(Œ481ÆÇH/z–œ(44 in
† ƒ}• ˆ ofƒ 4) we want‘ to find the distance from an element /0«”1C4
to the set s . Write
«
Gp—™˜MY D I ÍÏÎ
/ 1
/0›]481 D 4 and C
Gp—c˜MY
q
/ž9 E I ž1 ÉÒÑÔÓ/0›]4a1 E 4
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THE STRUCTURED DISTANCE TO NEARLY NORMAL MATRICES.
‘
Let /0‡<1’“4Ï)
s with
Gg—™˜Y I
/r
9 1ø19Š4 and ’
‡
Gg—™˜Y
I ž1 ɹú`1%2 4 q
/¹9 2Ï
The distance between /0«”1C4 and /2‡1’+4 is
—™˜Y
7
ÍÎ
/ D 99r1
/0›]49'ø1
—™˜MY
h
9F9Š4k7 - X
Ê 7
/¹9/
D
E
9G2 4a1žÑÔÓ/0›]49Cú`1
h
9G2 457 - 1
E
which, by equations (4.1) and (4.2), is
ÍÏÎ
7
h
/0›]49Cø7 h Ì
Ê –r7
D
h
h
99A7 h Ê 78ÑÔÓ/2›]49CúŠ7 h Ê:–r7
h q
E
9H27 h
GJI<K
The first and third term are minimized by the choices of constant ! -tuples ø
GUI
°3S ±
K
MLNPV3W
and ú
minimizing ?
h
d
9G27 h 1 where 91X2&)v> ?.Ÿ satisfy (3.2).
G
"iG ! õ
G#"
d
$ . Recall that 9
ÑÔÓ/ ÷ d ù I d 4 and
Let 91%2&)v>?.Ÿ satisfy (3.2) and let –A7
(4.5)
h
99A7 h |
Ê –A7
°TS ±
?
s to /2«”1C4 reduces to
‘
, cf., [5]. Thus, finding a closest element in
MLNPORQ
D
E
Thus equation (4.5) is
–A7
D
ªÍÎ ¾
h
7 h 9
/
G
Now, 7Y27 hh
d
 _.? n Ÿ d
D
–r7
(4.6)
D
Äù
! 1
"
2
‹ 4Ê|–
Ä GZ
d h
! [Y
2
.
Ä "BÄ h
ª ÍÏÎ ¾ E
h
h
h q
7Y27 h Ì
Ê –r7 E 7 h 9
/
1X2 ‹ 4cÊ|–A727 h
Ä "BÄ G ! õ
$ í . Therefore, equation (4.5) is
, and h
©
Ä "Ä h –
h
h
7h Ì
Ê –r7 E 7 h ÊX/ Ê
4
:ŒÄ ù
Äh
d
^ d
9
ªÍÏÎ ¾
/
"I D
Ê
E
1%2 ‹ 4
q
¿ G
ù
[
481 where we can
7
!
Ê
M/ \ _ 4 and recall that 2 has the form / _ 4
/
Let \
GB(˜M\
ÍÏÎ
M/ \ž_½4 . That is, we fix 5_ so that
*/ \ž]
_ ªŸ 4 is maximal. Thus,
choose 5_
G
"I D
E
G
Ä Äº ñ
d
_ G
ÍÎ ¾
G ?.l Ÿ Ä Ä ù d
¿
/ ½\ 1X2 ‹ 4
\¹_
^

d
_n d
d
Ä
ĺ ñ q
d .? l Ÿ Ä " I D
_ Ê E _
=
_
^ d _n d
e
Ä Ä
õ
õ
The above calculations lead us to define the function ×}*/ œ1 ù d 4 from §X"v§
to §
by
(4.7) ×F/Mœ1
Äù
Ä G
d 4
–
:
Ä Ä
/ ù d  |
Ê ^ hd Ê0 h 4
Äù Äh
Ê
^d d
¿
Äù
.? l Ÿ d º ñ Ä
Ä Äh Äq
_ z/ ^ d 9'Ék4 D _ Ê ù d E _
Äù Ä¿
d ^ed _ n d
9

lemma indicates,
finding the distance between a given pair /2«”1C
ƒ following
m
m 4 in
† ƒ • Asˆ the
‘
m
and s is equivalent to minimizing the above function.
ñ G
GâWY‰ZŠ\
_
 n d ^ and
L EMMA 4.2. Let V
/z^ 4 satisfy condition (1.1), and define _
•
ƒ
ƒ
ˆ
†
:ýG
d ñ
 _?.n Ÿ d _ . Let /2«”1 C4”)
with
Gp—c˜MY D I
Gp—™˜MY E I
q
«
/ 1 ^ 1 D 4 and C
/¹9 1ž`É _31 E 4
Ä Ä
Define ×F*/ ]1 ù d 4 as in (4.7) and let
õ q
GgY‰Þß5t
Ä Ä P
Äù Ä
×
×FM/ œ1 ù d 4
*/ ]1
d 4%)v§"v§ w
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108
LAURA SMITHIES
_? n dba , and  _? n d
d c , respectively. Then,
?
?
Let ø 6 and ú 6 be the constant ! -tuples with entries 
‘
the distance from /0«”1C4 to s is
‘
G
h
, - / /0«”1C481 s 4
(4.8)
h
h
7^9 
ø6 7 h *
Ê 7Y_v9 úœ6 7 h Ê̖r7
q
h
h
7h Ì
Ê –r7 E 7 h ÊÌ×
D
Moreover, this distance is bounded above by
‘
, h- //0«”1C481 sc4deS7^9 
ø 6 7 hh Ê7Y_9 úœ6 7 hh Ê̖r7
Ä
Ä
D
?.l Ÿ d º ñ Ä D
 :
7 hh Ê̖r7 E 7 hh Ê / 9
_ _ Ê
_n d
"G
õ
!
G
E
Ä q
_ 4
GB(˜M\
I D _]Ê E _ , and ½_
$ , \ž_
Proof. Fix /Mœ1 ù d 4 and define the variables
/M\¹_½4 ,
GS©
©
‘
Äù Ä
1kjkj5j1ž!9 . Let /2‡1M’“4”) s be given by the /y!ØÊ0ª4 -tuple /0ø1Múœ1œ1 d 1½Ä /@
_ 4žÄ 4 .
for all ð
We saw in the discussion above, these choices of ø“ú , and /@
_ 4 are optimal for this /*]1 ù d 4 ,
h
and , - /ž/2«”1C4a15/0‡<1’+4ž4 is given by (4.8). The upper bound is given by noting that the last
GâY‰Þß5t
Ä Ä P
Ä Ä
õ
×F/Mœ1 ù d 4
/*œ1 ù d 4“)̧²"#§ w , in the distance formula is less than or
term, ×
equal to
"
d
Äq
o º
Gp?:
 ?.l Ÿ º ñ Ä D
×F/ 1 e
^ dk4
9
_ _%Ê E _
_n d
Ä Ä
õ
The difficulty of minimizing ×}*/ œ1 ù G d 4 over §&"ç§
depends, of course, on the specific
/zU1OUA4 ,
matrices /0«”1C4 . For example, if /0«”1 C<4
: Ä
Ä
Ä Ä G
Ä Ä G – / ù d  ÊÌ^ hd Ê= h 4 q
Ä Ä
×FM/ œ1 ù d 4
× ;x/ ù d 4
^ d ù d h
GXo Ä Ä h G
 :
?
We have seen that this is minimized at m , ù d
^ed and the minimum is .
m
m
The above results translate directly into distance formulas for .
ñ
G
G²WY‰ZŠ\
_
 n d ^
C OROLLARY 4.3. Let V
/2^ 4 satisfy condition (1.1), and define _
ñ
: G
Gڗc˜MY
d
 _?.n Ÿ d _ . Let (
/ šX1 f1žr4 . Define f 6 to be the constant ! -tuple with entry
and
I K
G
i
hPj °‰êA±
MLNPg , and define
?
?
Ä
Ä Ä
Ä Ä
Ä ñ
: Äù
Ä
?.l Ÿ d /2^ed%9 ù d h 9'Ék4¹š_=ÊX/2^ed=Ê ù d h 9#Ék4¹O_ º _ q
Äù Ä G
/ d  Ê|^ hd Ê0 h 4
›M/ œ1 d 4
Ä Ä
9
Äù Ä¿
–Š^ed ù d h
d ^ed
_n d
Let
k
GpY‰Þßkt
õ q
Ä Ä
P
Ä Ä
›/ž/*]1 ù d 4ž4
/Mœ1 ù d 4%)§X"v§ w
Then
k
G
h
, - y/ (H12$ s 4
In particular, for any (*)
its distance from $ s is bounded by
o º
G
, - /y(1z$ s 4de:›// 1 ^ed54ž4
h
q
h
h
h
7f9 f
Ê 7kš37 h Ê78c7 h Ê
6 7 h X
h
h
h
7fØ9 f6 7 h ÊX7kš37 h Ê78c7 h Ê
:
9
.? l Ÿ d
_n d
Ä Äº ñ q
– O _
_
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THE STRUCTURED DISTANCE TO NEARLY NORMAL MATRICES.
G
Proof. /0Æ È /2–](481MÆ Ç /z–œ(44
—c˜MY
/ šÊT1O–
ÍÏÎ
/lf481šÊFr481
—c˜MY
/ š9#1O–œÉÒÑÔÓvl/ f481ž9vš4ž4 .
/0Æ È z/ –œ(4a1Æ Ç /2–](4ž4 in the previous lemma, then equation (4.8) is
Gp


h
h
h
h
h
h
7k–f9T– f6 7 h Ê̖r7kšçÊýc7 h Ê|–A78+9'š37 h Êý×
7f9 f36 7 h Ê 7aš37 h Ê 7aš37 h Ê|×1
Ä Ä

Ä Ä
G
E G
š}Êp and
}9&š . This,
and the function ×F/Mœ1 ù d 4 reduces to ›/*]1 ù d 4 for D
G
‘
d , h /ž/0Æ È /2–](481MÆ Ç /z–œ(4481 s4ž4 , establishes the
h
combined with the fact that , - /y(1z$es4

If /0«”1C4
G
/
109
corollary.
hPj °Åꁱ ®
G
(g9
Let ( )# . Define ( ;
. Then ( ; is the translation of ( , by a multiple of
the identity matrix, which has minimal? Frobenius norm among all such translations, cf., [5].
The above bound for the squared Frobenius distance from ( to $ s is less than or equal to
GXo
(with equality holding if and only if 
) the sum of the squared Frobenius distances from
(; to U and from U to $ s . We will see the importance of the translate, ([; , in the next section.
5. Applications. In this section, we bound the Frobenius distance from normality for
the elements of $es . We also apply results from [2] to indicate how well the set of eigenvalues
of an element of $s can be approximated by using normal matrices and Geršgorin-type sets.
First, let us consider what the above results tell us about the Frobenius distance† from
• ˆ
G
normality for the elements of $ s . Let (Ö)'>?A@? . The direct sum structure >?A@B?
and the Pythagorean relationship (2.2) imply that
‚
ˆ
†
YÅÞct h
Y‰Þt
h
h
h
h q
, - 0/ (1 4meÌÓ
, - /y(1 481M, - /0(1 48wneÌÓ
7kÆHÇ/y(457 - 1½7kÆÈ/y(457 - w
Recall that we defined
—™˜
® q
/y(4 A
(:9
!
‚
‚
G h
°‰êA± ®
h
is scalar, , - /y(H1 4
, - /y
( ;ª1 4 , cf., [5, Theorem 3.2]. ThereG
(;
Because the matrix
fore,
hPj
?
‚
YÅÞt
h
h
h q
, - 0/ (1 4de:Ó
7aÆÇH/0(o;`4k7 - 1½7aÆHÈ/y(;½457 - w
܁(*)> ?A@B?
Now let DŽ)ç$ s . By Corollary 3.5,
©
–
—c˜MY
"I
// 9
©
©
—™˜Y " I
©
o "
©
4 Ï
2 1øiÊýÉÔúœ15/ Ê
`4 2 4a1 and D ;
/ž/ 9 4 2”1 1½/ Ê 4`2 481
–
¿ "G ! õ
G
G d —™˜Y " I I o "
G
7 ! 45T6 . Then ÆHÈ/2D;`4
$ and ù _
where
/ 21 1 2 4 and ÆHnj/2D;`4
h
!
d —™˜Y /¹9 2
I 1 o 1X2 4 . By equation (4.2), 7aÆ È /zD ; 457 h G –A7 p 7 h G ° ! õ $ ± Z , and 7aÆ Ç /zD ; 457 h G
h
h h
h G Z
q
Äù Ä
d
h
for DE)+$ s defined by the /0!<G
Ê x4 -tuple, /2ø1Mú`1œ1 d 15/*
k_½4 _?.n Ÿ d 4 , the
–r7 p h 7 h
h [! Y Thus,
‚
distance from D to satisfies
:
:Ä ù
Ä
‚
YÅÞt /2^ hd 0
Ê h 4
d h q
h
Ä Ä 1
, - /zD*1 m
4 e:Ó
w
–ª^d ù d h
–Š^ed
Ä Ä
Graphically, this bound can be described as follows. Let DE)+$cs be given by /0ø1Mú`1 œ1 ù d 1
d
@/ _ 4 _ ?n Ÿ d 4 . The Frobenius distance from D
to the set of normal matrices is determined by
Ä Ä
G h
» Ê:^ hd . Specifically,
where the ordered pair /*œ1 ù d 4 lies in relation to the parabola ¼
qrr
YÅßÙÄ ù Ä 
û
°!! õ $ ± Z
rr
d ‹ h ÊÌ^ hd
rr
h D
G
‚
h
, - 2/ D*1 4me
©
"
s
rrt
rr
rr
h ! Z
G
°!! õ $ ± Z
h G
h ! Z
YÅßÙÄ ù
Ä G
d 
YÅßÙÄ ù
Ä ¾
d 
h ÊÌ^ h û
d
h ÊÌ^ h q
d
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LAURA SMITHIES
Note that {~s , the set of elements from $s with minimal Frobenius norm, corresponds
‚
G h
:
» Êý^ hd . For every D s ){~s , the above bound says , h- /2D s 1 4de
to the vertex of ¼
.
This is, of course, consistent with Theorem 4.1 which tells us that the normal matrix U has
:
distance from D s .
We now use the results of [2] to describe the distance from normality, in the sense of the
numerical stability of eigenvalue estimation through normal matrices, for the elements of $ s .
Recall that a singular value decomposition (SVD), of a non-zero complex !}"þ! matrix ’ , is
an expression of ’ as a product
’
(5.1)
Gvuw
G
¡
Ä £
jkj5j
jkj5jŽ¨ Ä ? ¦
jkj5j
Ä
¢ 
¨ Äd
u
o
£¥¤ ¡
o
¦
o
š d
¢ o
o
..
o
.
w
9
¢
š ?
9
..
.
x
I ê
d
x
I .ê
?
..
£¥¤
9
9
..
. ¦ 1
where and
are unitary matrices in >?A@B? , and is a nonnegative
diagonal matrix. The
w
Ä Ä
entries of , šedFf¸šrh#fÚj5jkj%f¸š , are the eigenvalues of ’ arranged in non-increasing
?
order. They are called the singular values of ’ .
Gyuw
Fix any non-zero ’·)|> ?@? and a SVD, ’
, as in (5.1). In [2], we defined
uw
the SV-normally estimated Geršgorin set, z|{}~/
. Like the Geršgorin set for ’ , the
4
u<w
set z{[}Y~/
4 is a union of closed discs and it contains the eigenvalues of ’ . We also
defined the SV-normal estimator \€‚oƒ…„ corresponding to this SVD of ’ . Specifically, define
GS©
m m
m
for each ò
1kjkj5jem 1ž! ,
\
GSº ©
9
Ä ¾
¨ 1
x
‹
Äh
o
and let \€oƒ„
m
G
dӈ
©
Z‡† t
ˆ
?
\
w
q
The parameter \€‰ƒ…„ lies between and , inclusively. It is used as a type of condition
uw
number which indicates how well the set z|{}~/
4 estimates the eigenvalues of ’ .
uw
When \€‰ƒ…„ is zero, z{[}Y~/
eigenvalues of ’ ; when \]€‰ƒ…„
4 is exactly the setuof
w
4 provides a gooduestimate
is small, the centers of the discs which comprise z‚{[}~Ï/
of
w
m m m
4
the spectrum of ’ . This is because the radii of the discs which comprise z-{}~”/
are
G º
–  š h \ h . Roughly speaking, up to a scaling factor of š d , this common radius
all Š
will be small when \ €‚oƒ „ is.
Finally, we cite the following lemma which bounds the SV-normal estimators from below. Notice that this lower bound on \ €‚oƒ „ is independent of the choice of SVD for ’ .
L EMMA 5.1. (See [2]) Let ’€)}>-‹@Œ‹ and let \€‚oƒ…„ denote the SV-normal estimator
Guw
corresponding to a SVD, ’
. Then,
7k’
’S9'’“’
h
7h ´
e
7a’7 -
\ €oƒ
„
q
The above lemma allows us to describe how well the spectrum of an element of $cs can
u<w
be approximated with the SV-normally estimated Geršgorin set, z|{[}Y~/ Ä Ä 4 .
m
m )ç$ s be defined by the /0!“ÊŽª4 -tuple /0ø1Oú`1œ1 ù d 1½/@
5_½4 _?n Ÿ d d 4 , and
T HEOREM 5.2. Let DŽ
:|G
d
_
 _?.n Ÿ d  n d ^ . Define
recall that
ä
Let D
\ €‰ƒ…„
Gu<w
. Then
G
!3/2ø h Ì
Ê úh 4

Ê
be a SVD of D
Ä Äh q
^ hd 0
Ê h
/ Äù Äh Ê ù d 4
–Š^ed
d
:
and denote the corresponding SV-normal estimator by
7aV7 h
ä
e\Y€oƒ
„
q
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THE STRUCTURED DISTANCE TO NEARLY NORMAL MATRICES.
In particular, if DŽ)v{ôs then
7aV7 h
:
Proof. Let
G
D
7kD
DÐ9TDSD
G
Ä
–
—™˜Y
q
"
©
"
©
/ž/ I 9 4 ”
2 1MøŒÊýɹú`15/ Ê
42 481
vG
d
d 1½/@
5_½4 _?.n Ÿ d 4 . Since D )Ì$ s , D DŽ9pDSD
7kh
7kVŒ7kh . We showed in Section 4 that
"
©
"
©
h
h
/ I 9 4 2I h
/ Ê 42 h q
h G !3/2ø  ÊÌú 4
75D¸7 ÊX7
7 h Ê7
7h
–
–
be defined by /2øA1Oú`1]1
Äù
©
e'\€oƒ…„
V and we have
Thus,
h G
7kD¸7 Since
Ä "BÄ h
Ê
Ä "I
.© Ä h
Ä"
x© Ä h
!3/0ø h Ì
Ê úh 4
9
Ê
Ê
h G

ÊX/

k4 7Y27 h
©HG
!
õ
$í
©
Ê
G
75D¸7 hFinally, if Du){
/
Ä "BÄ h
–
Ê
©
4
h q
727 h
GiZ !
[
,
and 727 hh
!3/2ø h :
Ê úh 4

Ê
Ä Äh G
q
^ hd 0
Ê h
ä
/ Äù Äh Ê ù d 4
d
–ª^d
G :
h v
:
s , by Theorem 4.1, 7kD¸7 7kD
!3/0ø h Ì
Ê úh 4

Ê
DÐ9TDSD
7kD¸7 h-
7kh G
. Thus,
7kVŒ7kh
:
e‘\ €‰ƒ
„ª1
in this case.
The previous theorem has an interesting interpretation. The elements of {·s have minimal Frobenius norm. However, the square of the reciprocal of this Frobenius norm is a factor
of our lower bound for \ €oƒ „ . Consequently, the condition number of the SV-normally
o
estimated Geršgorin sets for elements of {;s is maximally bounded above . This seems
to suggest the ucounter-intuitive
idea that, regardless of which SVD is used, the radii of
w
the set z{}~/
4 should tend to be largest for smallest elements of $ s . However,
h
this suggestion fails to consider how weighting factors š _ of the radius Š increase with
G
 _? n d š _ h .
7kD¸7 h6. Conclusions and extensions. We conclude this note with a few remarks and some
indications of further lines of inquiry for the sets $s . As we mentioned in Section 3, the right
G
D 9ýDSD
V has to be self adjoint with trace zero,
hand side of the matrix equation D
V
since the left hand
side
is.
The
choice
to
make
diagonal
arose from a desire to simplify the
‘
calculations for s and $ s and to make the rank of V easy to identify, since this rank is a type
of measure of the extent to which the matrix D fails to be normal. It would be interesting to
consider how the above development changes for the general right hand side, V .
‘
The intermediate set s was used to simplify the calculations for $ s and to help clarify
how the upper and lower bands of the elements of $ s are related to each other. However, the
‘
set s has an interesting
intrinsic functional analytic structure. Specifically, the decomposiG † • ˆ
tion >?A@?
expresses >?A@B? as a Kreı̌n space. This is an indefinite inner product
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LAURA SMITHIES
space which has the structure of the direct sum of a Hilbert space and a negative Hilbert space.
The structure of the operators on such spaces have been studied in detail, cf., [1,
‘ 4], and an
interesting line of inquiry would be to examine the properties of the elements of s as Kreı̌n
space operators.
Finally, other SVD-based Geršgorin-type sets were developed in [6] and [7]. The difficulty in applying such methods generally arises from the nonuniqueness of SVDs. We see
from the above development that elements of the algebraic varieties { s and $ s have sufficiently strong structure constraints to overcome the difficulties created by the nonuniqueness
of the SVD for normally estimated Geršgorin sets of [2]. An interesting question is whether
this happens with other SVD Geršgorin-type sets.
REFERENCES
[1] J. B OGNAR , Indefinite Inner Product Spaces, Springer, Berlin, 1974.
[2] N. F ONTES , J. K OVER , L. S MITHIES , AND R. S. VARGA , Singular value decomposition normally estimated
Geršgorin sets, Electron. Trans. Numer. Anal., 26, (2007), pp. 320–329.
http://etna.math.kent.edu/vol.26.2007/pp320-329.pdf
[3] R. H ORN AND C. J OHNSON , Topics in Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge,
England, 1994.
[4] H. L ANGER , Spectral functions of definitizable operators in Kreı̌n spaces, in Lecture Notes in Mathematics,
vol. 948, Springer, Berlin, 1982, pp. 1–46.
[5] S. N OSCHESE , L. PASQUINI , AND L. R EICHEL , The structured distance to normality of an irreducible real
tridiagonal matrix, Electron. Trans. Numer. Anal., 28 (2007-08), pp. 65–77.
http://etna.math.kent.edu/vol.28.2007-2008/pp65-77.dir/pp65-77.pdf
[6] L. S MITHIES , A note on polar decomposition based Geršgorin-type sets, Linear Algebra Appl., 429 (2008),
pp. 2623–2627.
[7] L. S MITHIES AND R. S. VARGA , Singular value decomposition Geršgorin sets, Linear Algebra Appl., 417
(2005), pp. 370–380.
[8] R. S. VARGA , Geršgorin and His Circles, Springer, Heidelberg, 2004.