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Minimal Geršgorin sets and ω-matricesG. M. Engel a & R. S. Varga ba International Business Machines , 70 IBM 68, Owego, N.Y., 13827b Department of Mathematics , Kent State University , Kent, OH,44242Published online: 02 Apr 2008.

To cite this article: G. M. Engel & R. S. Varga (1977) Minimal Geršgorin sets and ω-matrices, Linearand Multilinear Algebra, 5:1, 1-10, DOI: 10.1080/03081087708817168

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Linear and Multilinear Abebra, 1977, Vol. 5, pp. 1-10 0 Gordon and Breach Science Publiiers Ltd., 1977 Printed in Great Britain

Minimal Gersgorin Sets and a-Matrices

G. M. ENGEL? International Business Machines. 70 IBM 68, Owego, N.Y. 13827

and

R. S. VARGAf Department of Mathematics, Kent State University. Kent, OH 44242

Dedicated to Ky Fan, on the occasion of his sixty-second birthday, September 19,1976

(Received November 25, 1975)

If G(C) denotes the minimal GerSgorin set for C E Q="vn, and if, for any nonempty subset a of the first n positive integers, C[aJ denotes the principal minor of C determined by a, then conditions are determined which characterize matrices A and B in @"pn such that the inclusions G((D + B)[a]) G G((D + A)[a]) are valid for all subsets a of the fvst n positive integers, and for all diagonal matrices D in Cnsn. Connections with the newly defined set of w-matrices are also included.

1. INTRODUCTION

For any matrix A = [aiqj] E Pn, let G(A) denote its minimal GerZigorin set (cf. [63), i.e.,

n

[Z E @: - ai,rl < C IaijIxj/xiI , j - 1

] (1.1) jri

tResearch was made possible through the cooperation of JBM and the Department of Mathematics, State University of New York at Bhghamton.

SResearch supported in part by the Air Force Oflice of Scientific Research under Grant AFOSR-74-2729, and by the Energy Research and Development Administration (ERDA) under Grant E(l1-1)-2075.

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G. M. ENGEL AND-R. S. VARGA

where R: denotes the collection of column vectors x = [xi, x,, . . . , ~n]* in R" with xi > 0 for all 1 < i < n, and where the sum in (1.1) is defined to be zero when n = 1. With this notation, we seek conditions on matrices A = and B = [bjJ in Cn.* which insure that

{ G((D + B)[a]) E G((D + A)[a]) for all 4 # a E (n): = (1,2, . . . , n) and for all D = diag [dl, d,, . . . , d,] E a="*", . (1.2)

where A[a] in general denotes the principal submatrix of A determined by a, i.e., A[a] = where i, j e a. Our main result, Theorem 6, gives two conditions on A and B, each of which is equivalent with (1.2).

On considering (1.2), we &st observe from (1.1) that, on choosing a = ( i ) for any i E (n), the inclusion of (1.2) implies that

= bi,i, for all i E (n). (1.3) Next, we also observe from (1.1) that the off-diagonal entries of A enter into the definition of the minimal GerSgorin set, G(A), only through their moduli. This suggests that conditions which insure (1.2) will similarly depend only on the moduli of the off-diagonal entries of A and B.

2. NOTATION AND TERMINOLOGY

In this section, we collect some needed notation, terminology, and back- ground material on minimal Gersgorin sets and w-matrices. To begin, assumed that A = [aisj] E @"I" is reducible (cf. [5, p. 461). It is well known that there is a permutation matrix P E R"'" for which

where each Aj,j, 1 G j 6 r, is irreducible. Here, for convenience, all 1 x I matrices are defined to be irreducible. Then, it is known (cf. [6, p. 7251) that the minimal Gersgorin set for A is precisely the union of the minimal GerSgorin sets for Ajj, 1 ,< j < r, i.e.,

G(A) = 6 G(Ai,,). i = l

(2.2)

Next, consider any real matrix A = [aivj] e R"+ satisfying aiei real; ai,j 0, i # j ; for all i, j~ (n}. (2.3)

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Such matrices are called essentially nonnegative (cf. [I], [5l). As a consequence of the Perron-Frobenius theory of nonnegative matrices, such a matrix satisfying (2.3) possesses a real eigenvalue v, i.e., v E spec (A): = {A: det (A - U ) = 0}, which satisfies

' v >, Re A, for all A E spec (A), (2.4) and the inclusions

min { xj/xi) < v < max { x ai,j xj/xi), for all x E BB;. ir(n> je(n> if(n> j d n >

(2.5) Furthermore, v is characterized (cf. [3, p. 2011, [4]) by

v = inf { max [ C ai,j xj/xJ>, xeP+ ie(n> je(n>

(2.6)

and, if A is irreducible,

v = C at,,. xj/xi for all i E (n), for some x E 58:. (2.7) A < n >

These facts can be used as follows. For any A = [aivj] E Cnsn and for any z E @, define the real matrix T(z) = [ti,j(z)] E Rn*" by

t i , = - 1 - , I t i , j ( ~ ) = lai,jl, i # j ; for d l i, j~ (n), (2.8) so that T(z) satisfies (2.3). Denoting the associated eigenvalue of T(z), satisfying (2.4)-(2.6), by v(z; A), then v(z; A) satisfies, from (2.5) and (2.6),

min { - lz - ai,J + ]ai,j]xj/xi) < v(z; A) ie(n> &<?>

lit&

' 4 rnax { - Jz - ai,i/ + C l ~ , , ~ J x ~ / x ~ ) , for all x E R:, W n > &<n>

j#i

and V(Z; A) = inf { max [-(Z - aiil + x lai,jlxj/xi]>- (2.10)

XEW) k ( n > j+> j# i

As proved in [6], the points of G(A) can then be characterized in terms of the function v(z; A) by means of

PROPOSITION 1 For any A Fn, z E G(A) z$v(z; A) >, 0. We also include from [6] the following result.

PROPOSITION 2 For any irreducible A = [ai,.] E @"'", then z E aG(A), i.e., z is a bou?zdary point of G(A), implies that there exists an x E R",uch that

lz - ai,il = ]ai,,lxj/xi, for all i E (n). 72:) Next, following Engel and Schneider [2], a matrix A E Cn3" is called an

o-matrix if spec (A[a]) n R # 4, for all a with 4 # a E (n), (2.12)

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4 G. M. ENGEL AND R. S. VARGA

and, on dehing l(A[a]) : = min{spec(A[a]) n R), (2.13)

if, for arbitrary subsets a and j3 of (n), one has the nesting property

4 # a G B G (n) implies I(A@]) ,< l(A[a]). (2.14)

If wcl) denotes the set of all @matrices in Cn+", then, as stated in [2], w(,) contams all the Hermitian matrices, M-matrices, as well as all totally non- negative matrices, in @"*".

Continuing, (cf. [2]), a reflexive and transitive relation can be defined on o(,> as follows: if A, B E a(,,), then A 2, B if

I(A[al) B I(B[al), for all a with # a E (n). (2.15)

In the case that A and B are Hermitian matrices in Cnv", and thus elements of a(,), it is easy to verify that A 2, 0 iff A is positive semi-definite, and that A 3, B if A - B is positive semi-defhite. In the same vein, we investi- gate implications of the relation A 2, B on a subset, Bn", of w(,) which contains the M-matrices of o(,). Defining F'" as the subset of WF of all real matrices A = [aiJ] for which

aiPi real; < 0, i z j; for all i, j e (n>, (2.19

then from (2.3), A E Zn*" i£€ -A is essentially nonnegative. Thus, applying the characterizations of (24)-(2.7), it analogously follows that A = [atJ] E

Fen has a real eigenvalue I(A) such that I(A) ,< Re A, for all A E spec (A), (2.17)

min { C ai,j xj/xi) 4 I(A) ,< max { x aitj xj/xi), for all x e R;, W n > j€<n> ie(n) je(n)

(2.18)

I(A) = sup { min [ x ai,j xj/xil), (2.19) xcRq ie(n) I+>

and if A E P*" is irreducible, then I(A) = aiPj xj/xi for all i E (n), for some x e R;. (2.20)

&<n)

It is not difficult to verify (cf. [5, p. 301) that l(A), so defined, also satisfies the nesting property of (2.14), so that Bn*" is a subset of w(,). This brings us to the following useful characterization of "2," on P. PROPOS~ON 3 Given A = [ai,j] and B = [bi,j] in P*", then A 2, B IT, for every x and y in R-dfor every 4 # a s (n), there exist integers i and k in a such that

Proof First, assume that A >, B. It su£Eces to consider only the case a = (n), since the proof for any nonempty a in (n) is similar. By hypothesis,

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I(A) > I(B) and hence, for any x and any y in R;, we have from (2.18) that

which directly implies (2.21). Conversely, it follows from (2.21) that

for every x and y in R;, and every 4 # u c (n). Hence, taking the supremum of the right side over x in R: gives, from (2. lg),

akPj yj/yk 2 I(B[u]) for some k = k(2) E a, for all y E BB",. j=

If A[a] is irreducible, we can, from (2.20), choose y E R; so that C yj/yk = I(A[u]) for all k E a, whence I(A[a]) > I(B[or]). j=

If A[u] is however reducible, there is a subset j3 with g5 # fl c a for which A@] is irreducible, and for which I(ADI) = I(A[a]). Thus, with the above irreducible case,

J(A[al) = I(A@1) 2 l(BDI) 2 l(B[ul), (2.22)

the last inequality following from the nesting property of (2.14), and (ii) implies (i). a

3. MAIN RESULT

We begin with

DEFINITION 4 Given A = [ai,j] and B = [bi,J] in Cnsn, then IAl dominates IBI for every g5 # u G (n) for which Bra] is irreducible, and for every x and every y in R;, there is an i E u fir which

C Ia,,jI~j/~i 2 C Ibi,jIxjIxi. jw , j=

(3.1)

Note that (3.1) of Dehition 4 is a condition like that of (2.21) of Proposi- tion 3, but differs essentially in that (3.1) holds for the same i in the sums of (3. l), while (2.21) holds for possibly different i and k in the sums of (2.21).

We note from Definition 4 that IAI dominates IBI iff, for any non-singular diagonal matrices S = diag[sl, s2, . . . , s,] and T = diag[t,, t , , . . . , t,], then IS-'AS1 dominates IFIBTI. As another characterization in terms of @matrices, we have

PROPOSITION 5 Given A = and B = [biVj] in Cnsn, then IAI dominates IBI D - IBI ar D - IAI for all real D = diag[d,, d2, . . . , d,] E R".".

Proof Assuming first that JAl dominates IBI, consider any real D = diag[d,, d,, . . . , d,] in Rn3" and any nonempty a c (n). If B[u] is irreducible,

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6 G. M. ENGEL AND R. S. VARGA

then, for every x and y in R;, it follows from (3.1) of Dewtion 4 that there is an i E a for which

Next, we note that D - IAl and D - IBI are elements of Z"pn and, since B[a] is irreducible, so is (D - IBl)[a]. Thus, from (2.20), we choose x E R; so that

dk - C Ibk,jl xj/xk = l((D - IBI)[aI), for all k e a. j=

With this choice of x E R", (3.2) implies that I((D - IBl)[al) min {di - lai,jl yj/yi}, for all y E R,f .

iel i=

Then, the characterization of (2.19) directly yields

l((D - IBMal) 8 - IAl)[aD. (3.3) If B[a] is reducible, the argument used in (2.22) of Proposition 3 can be repeated to show that (3.3) again holds, whence D - IBI 8, D - IAl. Conversely, if D - IBI 8, D - IAl for any real D = diag[d,, d,, . . . , d,] E

R".", assumed that a is any nonempty subset of (n) with B[a] irreducible, so that (D - IBl)[a] is irreducible. By hypotheses, l((D - IBl)[a]) 8 l((D - IAl)[a]) for any real D = diag[d,, d,, . . . , d,]. For any x e R:, define the real numbers

lbi,jl xj/xi, for all i E a, di: = 1 J=

( 0 , for all i 6 a, and set a: = diag[di, d,, . . . , &] E Rn*". With the definition of (3.4) and the inclusions of (2.18), it follows that

di - Ibi,jl xj/xi = 0 = l((d - IB1)[a]) for all i E a. (3.5) 1-

On the other hand, by hypothesis and by (2.18),

1 - I 1 - I 2 i - I j / i for any i a j=

Y E R;. (3.6) Thus, combining the results of (3.5)-(3.6), there is an i E a for which

di - C lb~ ,~ l xjlxi 2 di - C lai,,I Y~IY~, J - j=

which implies that, for any x and y in W;, there is an i E a such that

i.e., [A[ dominates IBI. Given A and B in [Wn9", if there is a particular real diagonal matrix D E Rn*"

such that D' - IBI 8, D - IAl, then Proposition 5 provides no information as to whether or not IAl dominates IBI. In fact, it is possible to construct an

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example where D - IBI 2, D - IAl for a particular real diagonal matrix, but where IAI fails to dominate IBI. To show this, letting

it can be readily verified that D - I B1 2, D - I Al. On the other hand, it can be verified that p(lAl) = $2 = 1 A14 21 3 and that p(lBI) = 1.434 467, so that p(lB1) > p(lA1). This, however, implies from Proposition - . 7 in the next section that IAI cannot dominate IBI.

This brings us to our main result.

THEOREM 6 Given A = [aij] and B = [biSj] in Cn*", the following are equiva- lent:

i) G((D + B)[a]) c G((D + A)[a]) for any 5, # a E (n) and for any D = diag[dl, d,, . . . , dn[e C*";

ii) G((D + B)[aU c G((D + A)[a]) for any 5, # a E (n) and for any D = diag[dl, d,, . . . , d,] e CnG such that (di + bi,i) is real for all i E (n);

iii) /A] dominates IBI and ai,i = bi,, for aN i E (n); iv) aiSi = bini for all i e (n) and D - 1B1 2, D - IAI for any real

D = diagrd,, d2, . . . , d,] E Rn*".

Proof That (i) implies (ii) is obvious. Assuming (ii), we fist observe, on choosing a = (i) for any i E (n), that (ii) implies from (1.1) that di + b,,i = di + aiSi for any di such that di + bi,i is real, whence ai,i = biSi for all i E (n). Next, consider any a with 5, # a E (n) for which B[a] is irreducible, and any x E R:. Now, define the numbers di so that

which implies that b: = diag[a,, a,, . . . , an] satisfies the hypothesis of (ii). With this choice of the &'s, it follows that

Iz - (ai + biJ = C Ibi,jl xj/xi, for all i e or, i€a

(3.8) > # i

is evidently satisfied for z = 0. Using the inclusions of (2.9), we see that (3.8) with z = 0 implies that v(0; (D + B)[a]) = 0, so that, from Proposition 1,0 E G((0 t- B)[a]). Thus, hypothesis (ii) implies 0 E G((D + A)[a]), whence from Proposition 1, v(0; (b + A)[a]) 2 0. With the second inequality of (2.9), there is, for any y E R;, an i E cc for which

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8 G. M. ENGEL AND R. S. VARGA

But as ai,, = b,,, for all ic(n), the above inequality becomes, with the definition of (3.7), just

i.e., IAl dominates IBI, and (ii) implies (iii). That (iii) implies (iv) follows directly from Proposition 5.

Finally, we show that (iv) implies (i). Assuming (iv), consider any a with 4 # a c (n) and any D = diag[d,, d2, . . . ,A] E Cn+. To establish (i), it sufiices to show that if z E aG((D + B)[a]), then z E G((D + A)[a]). More- over, because of (2.2), we may assume that (D + B)[a] is irreducible. Thus, as z E aG((D + B)[a]), it follows from Proposition 2 that there is an x E IW; for which (cf. (2.11))

lz - (di + br,31 = C Ibr,jl xj/xi, for all i E (n). je<n> j + i

Next, with the definition of the real numbers di in (3.4), we see that the asso- ciated matrix 8: = &ag@,, a,, . . . , A] is real, and the hypothesis of (iv) gives us that 8 - IBI 2, 8 - IAI. Now, from Proposition 3, this implies that, for any y E IW:, there exist integers i and k in a such that

Hence, from the dewtion of the dj9s in (3.4) and the hypothesis that ai,i = b,, for all i E (n), the above inequality reduces to

Coupling this with (3.9) and again using the fact that a & , = b,, then yield

I + k

Thus, from (2.10), we deduce that v(z; (D + A)[al) Z 0, whence, from Proposition 1, z E G((D + A)[a]), i.e., (iv) implies (i)..

4. PROPERTIES ASSOCIATED WITH THE RELATION 'A' DOMINATES 'B'

We begin with any easy consequence of Dehition 4. For notation, let p(C) denote the spectral radius of any C E Pn, i.e., p(C): = max (IAI: A E

spec (C)h

P R o ~ o s ~ n o ~ 7 Given A = [ai,.] and B = [bi,j] in Fa, then iAl dominates IBI implies that

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Proof If (A( dominates IBI, then, from Proposition 5 with D = 0, we have that - IBI 2, - IAl, or equivalently, that

K-(IBI)bI) 3 I(-(IAI)E~l) forany 4 # a c (n). (4.2) But, from the Perron-Frobenius theory of nonnegative matrices, it is easy to see from (2.19) that I(-(IBl)[u]) = -p((lBl)[aD, and similarly I(-(IAl)[u]) = p((lAl)[ul), whence (4.2) implies (4.1). rn

For the next result, we write in the usual notation of nonnegative matrices that IAI 2 IBI if A = [ai,j] and B = [biPj] in Cn*" satisfy

laisj1 2 lbisj1 for a11 i, j~ (n). (4.3) P ~ o ~ o s r n o ~ 8 Given A = and B = [bi,,] in Cn>" satisfying (1.3), then IAI 2 IBI implies that IAl dominates (BI.

Proof If IAI 2 IBI, then for any nonempty u E (n), for any real D = diag[dl, d,, . . . , d,] E Rn.", and for any x E R t ,

di - C lbi,jl xj/xi 2 di - laiPjl xj/xi, for all i~ u. j- j-

(4.4)

Hence, since D - IAl and D - IBI are elements of P n , it follows from (4.4) and the characterization of (2.19) that

I((D - (BJ)[u]) 2 l((D - JAl)[ul) for all nonempty cr E (n), whence D - IBI 2, D - IAl for all real D = diag[dl, d,, . . . , d,] E Wn. Thus, applying Propositions 5, IAJ dominates (B(.

Finally, we remark that the converse of Propositions 7 is false. As a counterexample, consider

0 0 1 0 0.08 4 , A : = [o 0 11, 1.1: = E : ;].

1 1 0 By direct computation, it can be verified that (4.1) is valid in this case for all 4 # a c (3). However, ]A] does not dominate IBI. To see this, choose 62 = (3), and % = [0.30, 1, 1IT and y" = [0.9091,0.5001, 1IT in RJ,. Then, as can be verified, the inequality of (3.1) of Definition 4 fails for every i E (3). In the same vein, the converse of Proposition 8 is easily seen to be false, since, with

IAI dominates IB( and IB( dominates IAl, but IAl # IBI.

References [I] Garrett Birkhoff and Richard S. Varga, Reactor criticality and nonnegative matrices,

J. Soc. Indust. Appl. Math. 6 (1958), 354-377.

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10 0. M. ENGEL AND R. S. VARGA

(21 Gernat M. Engel and Hans Schneider, The Hadamard-Fischer inequality for a class of matrices defined by eigenvalue monotonicity, Linear and Multilinear Abebra (to appear).

{3] B. W. Levinger and R. S. Varga, Minimal Gerschgorin sets. 11, Pacific J. Math. 17 (1966), 199-210.

141 C. Odiard, Un corollaire du theOr&me de Pcrron-Frobmius, Rev. Fran~abe Informat. Recherche O&rationnelle, 5 (1971), S r . R-2,124-129. -

[S] Richard S. Varga, Matrix Iterative Ana&sir, Rentice-Hall, Inc., Englewood Cliffs, N.J., 1962.

[6] Richard S. Varga, Minimal Gerschgorin sets, Pacific J. Math. 15 (1965), 719-729.

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