ETNA
Electronic Transactions on Numerical Analysis.
Volume 8, 1999, pp. 15-20.
Copyright  1999, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
ON GERŠGORIN-TYPE PROBLEMS AND OVALS OF CASSINI
RICHARD S. VARGA
AND ALAN KRAUTSTENGL
Abstract. Recently, two Geršgorin-type matrix questions were raised. These are answered here, using ovals of
Cassini.
Key words. Geršgorin circle theorem, ovals of Cassini.
AMS subject classification. 15A18.
1. Introduction. Given a matrix
, its usual Geršgorin disks are given
by
.
0/ 21354 76 213548129 #;:<#= 4>
$
#
"
!
(1.1)
%'%+&)&-* (,
1 54 denotes the spectrum of , i.e.,
and, if ?
? 1354 A@BCD B is an eigenvalue of EGF >
the famous Geršgorin circle theorem [3] (or Horn and Johnson [4, p. 344] and [7, p. 16]) is
that
6 21354) M6 1543N
3
1
5
4
H
(1.2)
?
JILK
@O
P F JILK and the = row sums @ / 21354 F JILK of
=
At this moment, only
the diagonal @ entries
2
1
Q
4
=
6
(1.1) determine the Geršgorin disks
F JILK and the inclusion set 6 154 , of (1.2), in the
complex plane. Then, with the notation
R A@S9.>2T$>0U0UVUW> = F >
T = pieces of information are used to define the following set of matrices:
these
XY Z[\J]2
C ]2
P ^_
P and / 213ZG4) / 1Q4 for all : R N
1Z`4 , for each Z_ X , must also satisfy the inclusion of (1.2), and, with the
It is evident that ?
(1.3)
notation
(1.4)
e
? 1 X 4 ca bMd ? 1Z`4>
Received March 30, 1998. Accepted for publication August 13, 1998. Recommended by V. Mehrmann.
Institute for Computational Mathematics, Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242 (U.S.A.) (varga@msc.kent.edu). This research supported by National Science
Foundation Grant DMS-9707359.
Department of Mathematics, Case Western Reserve University. 10900 Euclid Ave., Cleveland, OH 44106
(U.S.A.) (axk66@po.cwru.edu).
15
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16
On a Geršgorin Problem and Ovals of Cassini
we have
? 1 X 4H 6 1354N
(1.5)
As we shall see, the inclusion in (1.5) is not always one of equality.
Recently, the following questions were asked of one of us:
be given?
Question 1. Can a precise description of
Question 2. When is it the case that
is a single closed disk (with a
nonempty interior) in the complex plane?
The point of this note is to answer the above questions.
We remark that the set
of (1.4) is, for
, in general larger than the minimal
Geršgorin set (cf. [8]) for any matrix , simply because the minimal Geršgorin set uses
more specific information about the matrix
in determining its minimal
Geršgorin set. Specifically, the associated set of matrices, for the minimal Geršgorin set of a
, is
given matrix
? 1X 4
;AJ
P iC 1X 4
1? X ? 4
=gf T
_hJ
P 5D Xkj`l Z[\J] $$ ] _ and ] for all : >mG R >
@
P F JILK , one is also given the moduli of
=
entries
i.e., in
addition
to knowing the @ diagonal
1
O
9
4
P
=
=
! nondiagonal entries F ,o,on %'&$* &)% ( , for a total of =p pieces of information. Though
all
(1.6)
it is not essential for this paper, we remark that it is evident from (1.3) and (1.6) that
? 1 X j`l 4qH ? 1 X 4 for any rC > =sf T$>
where
in (1.7) cannot in general hold for
tit can
f T , butthatinthetheequalities
= rofT , containment
be> =shown
all
case
there holds
(1.7)
(1.8)
X j`l H X >
so that
X j`l X >
so that
? 1 X j`l -4 ? 1 X 4
for any
uC p p N
2. On Question 1. We need some additional notations and results. Given the
matrix
(2.1)
Zt
TvwT
]MKW Kx]0Ki p p p >
] p Kx] p p
its associated oval of Cassini is defined as
y 31 ZG4 A@ Cz 1 k! ]0Ki K{4+1 |! ] p p 4 $#u ]MKW p U] p K F N
C > =;} T , we denote its ovals of Cassini by
Then, for a matrix
(2.2)
y M1354 \@ 1 k! P 24+1 |! . W .4 #;/ 1Q4^U / .1Q4 F > for all :t~ m in R N
€Jp  Kƒ‚ ovals of Cassini y 71Q4 associated with the matrix , and from a
There are p
(2.3)
well-known result of A. Brauer [1] (see also [4, p. 380]), we have that if
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17
Richard S. Varga and Alan Krautstengl
(2.4)
y 1Q4 y P .13543>
,„,on %…&$* &^% (
then
(2.5)
? 1Q4†H y 1Q4N
But, as the ovals of Cassini in (2.3) use only the information given by
it is also true from (1.3) that
@
P F JILK
and
@ / 21354 F IcK ,
? 1 X 4 H y 1Q4N
y ‡ 71543> :ˆt~ m , is a bounded and closed
We remark from (2.3) that each oval of Cassini
y 1Q4 in (2.4). Next, if ‰
(hence, compact)
set in the complex plane , as
is<
their
finite union
‹
Œ
‰ denotes its complement. The boundary
is a set in , then ‰ denote its closure
‰ and‰ ‰|Š . Š
of ‰ is defined, as usual, by O‰
1X 4
4 . first result, which sharpens the inclusion in (2.6), shows exactly how ? fills out
y 135Our
ŽAJ.
$ > =} T , then
T HEOREM 2.1. Given any
(2.6)
(2.7)
? 1 X )4  y 154-  y KW p 1543>
when
= ‹T$>
and
? 1 X )4 y 1354> when =;}Ž N
= rT , each matrix Z in X is, from (1.3), necessarily of the form
Proof. For
KW K '‘O’ / K013543 '‘ ( >
[
Z
K > “ p arbitrary real numbersN
p p
/ p 1Q4
(2.9)
with “
B
Z
1Z ! BO”•4^u– , so that from (2.9),
If is any eigenvalue of , then det
1Ki K ! B^4+13 p p ! B^4) / K01354^U / p 1543 € ‘ (˜— ‘™’+‚ N
(2.8)
Hence,
! B U p p ! B / KM1Q4^U / p 1354N
Bš y Ki 1Q4 . Since this
As (2.10) corresponds to the
case ofZ equality
in (2.3), we see that Ki  1354cp B
X
y p
y 154 in this
is true for
any eigenvalue
ofKW any
in and
since, from (2.4),
›
T
1
c
4
H
3
1
5
œ
4
1
5
4
=
X
y
y
case Ki 135, 4 then ?
point
 p
 K . Conversely, it is easily seen that
Z ineach
y
p
of  1 p 4^ is, for
suitable
choices
of
real
and
,
an
eigenvalue
of
some
(2.9),
so
“
“
X  y Ki p 1Q4)  y 1Q4 , the desired result of (2.7).
that ?
(2.10)
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On a Geršgorin Problem and Ovals of Cassini
To establish (2.8), first assume that
which has the partitioned form
Z[
(2.11)
=Y}_ , and consider a matrix ZhJ] in ,
Z iK K Z Ki p >
ž Z p p
where
Ki
'‘™ K ’ 7Ÿ  '‘ ( > – # Ÿ #s/ K013543>œ– # #/ 1Q4>
p
p p with

(2.12)
K
]2W cAW for all 9 # m #;= . Now, for any
with “ andŸ “ p arbitraryŸ|real
numbers,
and with
s
–
>
V
K
1
Q
„
4
s
–
>
/ p 1Q4„ , the entries of the block ZˆKi p can be
/
choices of and with
and 13ZG4
0
K
1
`
Z
4
/ and / p , in the first two rows of Z , equal those of .
chosen so that the row
sums,
=}[ , the row sums of the matrix Z p p of (2.11) can be chosen to be the
Similarly, because
Z
same as those in the remaining row sums of . Thus, by our construction, the matrix of
X
= ,
(2.11) is
an
element
of . (We remark that this construction fails to work in the case
1
5
^
4
u
–
M
/
¡
unless
). But from the partitioned form in (2.11), it is evident that
ZˆKW K ? 13ZG4) ? 1ZˆKW KM4 ? 13Z p p 4N
ŸM> > “ Ki> “ p in ZˆKi K in (2.12), it can be seen from (2.3), that,
However, from
the
parameters
W
K
1
5
4
y p , there are choices forZˆthese
forZˆeach
parametersŸ such that – is # an Ÿ eigenvalue
W
K
K
W
K
K
#¢/ KM154
of – # . In #›
other
words,
the eigenvalues
of
, on varying and with
1
5
4
W
K
1
5
4
/
p , fill out y p , where we note that the remaining eigenvalues of Z
and
Z p ) must still lie, from (2.6), in y 1354 . As this applies to all ovals of
(namely, those
of p :D
M
3
1
5
4
£
~ m , upon suitable permutations of the rows and columns of Z of
y
Cassini
,4)for
1
1
5
X
y 4 , for all =s}D .
(2.11), then ?
=  of (2.8), any matrix Z in X can be expressed as
For the remaining case
KW
‘™ K ¤
ŸM '‘ (
1 / K01354 ! Ÿ4 '
'‘™‘™¥ ’
1 / 1354 4
Z[  …‘™§
>
(2.14)
¦
1 / ¡ 1354 p!šp ¦ 4 …‘™¨ p ¡ ! ¡
(2.13)
where
– # Ÿ #s/ K013543>œ– # #/ p 1Q4> and – # ¦ #;/ ¡ 1Q4> and
@ “ Fi©
JILK are arbitrary real numbers.
y Ki p 1Q4 , i.e., (cf. (2.3)), let Now, choose any complex number in the oval of Cassini
(2.15)
satisfy
k! Ki K U |! p p #;/ K 154U / p 1Q4N
Ki K œ#r/ KM154 or ˆ! p p ª#r/ p 1Q4 is valid.
The above inequality implies that either Q !
ˆŸ7 ! '‘ Ki K ª#[/ K01354 , choose the parameters Ÿ and
Assuming,
that
K“ such thatwithout
K AofŸ generality,
i
K
œ
K
›
"! Kiloss
( , and similarly, choose the parameters and
and "!
(2.16)
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19
and |! p p ¡  …‘™¤ , where, from (2.16), Ÿ<U #Ž/ K 1Q4•U / p 1354 .
“ ¡ such that k! p p '9.>M9>+–™­¬ in ® , it can be verified that Z « u¯ , where
Then with the vector «
Ki
' ‘ K¤ ±ŽŸM '‘ ( >
° KœA
° p¡   '‘ § ±;±z 1 p/ ¡ p 1354 > 4 …‘ ¨ N
(2.17)
° ¦
!š¦
/ ¡ 13543²VT and “†³ “ ±;´ , we have ° ¡ h– . Thus, Z « « ,
Then, on setting ¦
Z
in© y KW p 154 is an eigenvalue of some Z in X .
and is an eigenvalue of . Hence, each
=
Consequently,
this construction
be applied to any point of any oval of
:t~ m , then ? 1 X for4) y 1354 can
y .1Qas4 with
Cassini
, which completes the proof of (2.8).
X
We remark that there is a way to enlarge the set , in the spirit of what has been done in
the treatment
of minimal Geršgorin sets (cf. [8]), which gives
for
=r} T , which
= \aT unified
=u}rinclusion
 , as in result
any
does not distinguish between the cases
and
Theorem
Richard S. Varga and Alan Krautstengl
2.1. To this end, set
X µ ZtAJ]2
]2
P ^›
and – #;/ 1Z`4 #s/ 1354
Y
X H X µ , and this of course implies
so that
(2.19)
? 1 X 4H ? 1 X µ 4N
(2.18)
for all
: R >
But from the definitions in (2.3), we also have
? 1 X µ 4H
ŽAJ T HEOREM 2.2. Given any
(2.21)
? 1 X µ 4)
(2.20)
= AT
y 1Q4N
$ with =;} T , then
y 1Q4N
? 1 X µ 4q y = 135‹4œT
= ›T
Proof. From Theorem 2.1 and the inclusions of (2.19) and (2.20), it is only necessary to
establish (2.21) in the case
, i.e., for
, it suffices to show that
. The proof of this is similar to the proof in the first part of Theorem 2.1. For
,
each matrix in is, from (2.18), of the form
y iK p 1354
Z
Xµ
KW
'‘O K ’ 7Ÿ  '‘ ( > with – # Ÿ #\/ K 1354>k– # # / 1Q4
p p
K“ > “ p arbitrary real numbers. p ,

(2.22)
B
Z
1Z ! BO”•4^u– implies that
If is any eigenvalue of , then det
13 KW K ! B^4+1 p p ! B)4)›Ÿ  € ‘ ( — ‘ ’ ‚ >
Z[
“ iK > “ p N
Ÿ –> / KV1Q4„ and –> / p 1Q4„ , and as “ K and
But as and run, respectively, through the intervals
Z in X µ fill
it is135evident
that the eigenvalues
of
the
matrices
“ p run through all realy numbers,
µ
i
K
1
Q
^
4
4
g
T
>
1
^
4
i
K
1
Q
•
4
=
X
y
y
p
p y 154 .
out (cf. (2.3)) the set
, i.e., in this case
?
so that
13 Ki K ! )B 421 p p ! B^4 _Ÿ >
and
for all real
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20
On a Geršgorin Problem and Ovals of Cassini
t¶ ·¸ > =} T
3. On Question 2. The answer to Question 2 is straight-forward, based on the results of
Section 2 and the following observations. Given a matrix
, then,
with the definitions of
of (1.1) and
of (2.3), it is easy to verify that
6 +1354
y .154
y 1Q4¹H 6 1354 6 1354 for all : >mº R with :»[~ m>
/ 1354) / 1354)u–$>
(3.1)
with equality
holding
only
if
W
1
Q
4
)
_
W
/
/ 1Q4 f –N
or if
and
6 1Q4 of (1.2) and y 1Q4 of (2.4), that
The set inclusions in (3.1) imply, with the definitions of
one always has
y 1354qH 6 1354N
(3.2)
›¼J
P ˆ
Question 2, it can be seen from Theorems 2.1 and 2.2 that, if
So,$to > answer
=} T , then
= uT> ? 1 X 4 is never a single closed disk with a nonempty interior.
a) for
= ½T$> ? 1 X µ 4 is a single closed disk with a nonempty interior only if
b) for
KW Kœ› p p and / K0154U / p 1Q4 f – .
=f T> ? 1 X 4) ? 1 X µ 4 is a simple closed disk with nonempty interior only
c) for
Ÿ7> in R with Ÿ ~ such that ¾0 ¾c¿+ ¿ and / ¾.1Q4U / ¿i1Q4 f – ,
if there are
y P .1354 lie in y ¾0 ¿1Q4 .
and all remaining ovals of Cassini
1 X 4 , in
4. Final Remarks. It was a surprise for us to see that the determination of ?
Theorem 2.1, was completely in terms of ovals of Cassini, a topic rarely seen in the current
research literature in linear algebra. In fact, on consulting many (16) books on linear algebra,
we could find only three books where ovals of Cassini are even mentioned: in an exercise
in Varga ([7, p. 22]), in a lengthier discussion in Horn and Johnson [4, pp. 380-381], and
in Marcus and Minc [5, p. 149]. But, in none of these books was it stated that the ovals of
Cassini are in general at least as good (cf. (3.2)) as the Geršgorin disks, in estimating the
spectrum
of a given matrix in
. This appears in Brauer’s paper [1] of 1947 (which
curiously does not mention Geršgorin’s earlier paper [3] of 1931.) What apparently was of
more interest in the literature is the fact that Brauer’s use of two rows at a time, to determine
an inclusion set for the eigenvalues of a given matrix, cannot in general be extended in the
same manner to rows at a time, for
, and explicit counterexamples to this are given
in [4, p. 382] and [5, p. 149]. However, a “correct” generalization of Brauer’s results, using
graph theory, is nicely given in Brualdi [2].
? 1Q4
À
$
À }g
REFERENCES
[1] A. B RAUER, Limits for the characteristic roots of a matrix II, Duke Math. J., 14 (1947), pp. 21–26.
[2] R. B RUALDI , Matrices, eigenvalues, and directed graphs, Lin. Multilin. Alg. 11 (1982), pp. 143–165.
[3] S. G ERSCHGORIN, Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk SSSR Ser. Mat., 7
(1931), pp. 749–754.
[4] R. A. H ORN AND C. R. J OHNSON, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
[5] M. M ARCUS AND M. M INC, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston,
1964.
[6] O. TAUSSKY, Bounds for characteristic roots of matrices, Duke Math. J. 15 (1948), pp. 1043-1044.
[7] R. S. VARGA, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
[8] R. S. VARGA, Minimal Gerschgorin Sets, Pacific J. Math, 15 (1965), pp. 719–729.
[9] X. Z HANG AND D. G U, A note on Brauer’s Theorem, Lin. Algebra. Appl. 196 (1994), pp. 163-174.