First, we review some important facts on the location of eigenvalues of matrices.

BLOCK NORMAL MATRICES AND GERSHGORIN-TYPE DISCS JAKUB KIERZKOWSKI AND ALICJA SMOKTUNOWICZ Abstract The block analogues of the theorems on inclusion regions for the eigenvalues of normal matrices are given By an inclusion region for a given matrix A we mean a region of the complex plane that contains at least one of the eigenvalues of A Some nonsingularity results for partitioned matrices are also presented Key words Normal matrices, Block matrices, Eigenvalues, Gershgorin s discs, Nonsingular matrices, Hermitian positive definite matrices, Strong square-sum criterion AMS subject classifications 15A12, 15A18 1 Introduction The purpose of this paper is to derive new inclusion regions of Gershgorin type for partitioned normal matrices; see Theorems 27 28 Theorem 27 improves the results of Meyer and Veselić; see [5, p 436] We also give the block versions of nonsingularity results; see Theorem 29 Theorem 29 is an analogue of Varga s Theorem 62 for strictly block diagonally dominant matrices; see [11, p 157] We review some of the standard facts on the location of eigenvalues of block matrices which are useful in error analysis of numerical algorithms and in study convergence of block iterative methods for solving linear systems of equations The set of all n-by-n matrices over C is denoted by M n We recall that a normal matrix A M n is any matrix satisfying AA = A A, where A is the conjugate transpose of a matrix A Normal matrices A can be Hermitian (A = A), skew- Hermitian (A = A), and unitary (A A = I n ), where I n is the n n identity matrix We consider only the spectral matrix norm and the second vector norm Throughout the paper, σ(a) denotes the set of all eigenvalues of A, ρ(a) = max{ λ : λ σ(a)} is the spectral radius of A and R(x,A) = x Ax is the Rayleigh quotient of x C n with x 2 = 1 First, we review some important facts on the location of eigenvalues of matrices Theorem 11 (Gershgorin, 1931) The eigenvalues of A M n lie in the union Received by the editors on July 14, 2010 Accepted for publication on October 5, 2011 Handling Editor: Richard Brualdi Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl Politechniki 1, Warsaw, 00-661 Poland (JKierzkowski@minipwedupl, ASmoktunowicz@minipwedupl) 1059

1060 J Kierzkowski and A Smoktunowicz n G i of the n discs G i = {z C: z a ii r i (A)}, (11) where r i (A) denotes the deleted absolute row sums of A, ie, r i (A) = a ij (12) j i However, some Gershgorin s discs may contain no eigenvalues In case of normal matrices, such situation does not occur Theorem 12 ([7]) Let A M n be normal Then each of the discs E i = {z C: z a ii s i (A)}, (13) where s i (A) = a ij 2, (14) j i must contain an eigenvalue of A Using the Cauchy-Schwartz inequality we get s i (A) r i (A) n 1 s i (A), i = 1,,n (15) Thus, E i G i for all i = 1,,n, so the Euclidean discs E i are smaller than the Gershgorin discs G i Unfortunately, some eigenvalues of the normal matrix A can be located outside the set n E i Theorem 13 For any λ C and n > 2, there exists a normal matrix A M n such that λ is an eigenvalue of A and λ a ii > s i (A) (i = 1,,n) Proof Let A = (λ n)i n + J, where J has each entry equal to 1 It is easily seen that σ(j) = {0,n} Therefore, σ(a) = {λ,λ n} From this it follows that λ a ii = n 1 > s i (A) = n 1 for all i This completes the proof It is well-known that if A M n is strictly diagonally dominant (ie, A satisfies the strong row-sum criterion: a ii > r i (A) for all i = 1,,n), then A is nonsingular; see, for example, [1] Some other interesting nonsingularity results were given in [1] Now we present a new nonsingularity result involving the radii s i (A)

Block Normal Matrices and Gershgorin-Type Discs 1061 Theorem 14 If A M n satisfies the strong square-sum criterion, ie, then A is nonsingular ω(a) = n s 2 i (A) 2 < 1, (16) a ii Moreover, if A, satisfying the strong square-sum criterion, is Hermitian and all of its diagonal elements a ii are positive, then A is positive definite We give a proof of this theorem in Section 2; see Theorem 29 It is known that the strong square-sum criterion and the strong row-sum criterion are sufficient conditions for the convergence of the Jacobi method for solving a linear system of equations Ax = b; see [4, p 359] Notice that these criteria are not equivalent Example 15 Let A 1 = 88 7 12 17 7 78 17 12 12 17 68 27 17 22 27 58, A 2 = 14 6 4 1 6 24 11 3 4 11 20 3 1 3 3 11 It is easily seen that A 1 is not strictly diagonally dominant but satisfies the strong square-sum criterion with ω(a 1 ) 08393 On the other hand, A 2 does not satisfy the strong square-sum criterion, since ω(a 2 ) 10806 However, A 2 is strictly diagonally dominant We extend these results to partitioned matrices in Section 2 Some of them can be derived from the following residual theorem; see, eg, [4, p 104] Theorem 16 If A M n is a normal matrix with eigenvalues λ 1,,λ n, then for any α C and any vector x C n with x 2 = 1 min λ i α Ax αx,,n 2 To illustrate the theory, we present numerical tests in Section 3 2 Inclusion theorems for block matrices Let the matrix A M n be partitioned into q q blocks A 11 A 12 A 1q A 21 A 22 A 2q A = [A ij ] =, (21) A q1 A q2 A qq

1062 J Kierzkowski and A Smoktunowicz where A i,j C ni,nj is the (i,j) block of A, {n 1,,n q } is a given set of positive integers, and n 1 + + n q = n First, we recall very useful estimations of the eigenvalues of partitioned matrices, frequently used in error analysis of numerical algorithms and in the study of convergence of iterative methods for solving large linear systems of equations Theorem 21 Assume that A M n is partitioned as in (21) If λ σ(a), then λ ρ(µ(a)) µ(a) 2, (22) where µ(a) is a matricial norm of A, defined as A 11 2 A 12 2 A 1q 2 A 21 2 A 22 2 A 2q 2 µ(a) = (23) A q1 2 A q2 2 A qq 2 Theorem 21 is a generalization of the Perron-Frobenius inequality and was first proved by Ostrowski; see [2], [6] and [10] If, additionally, the matrix A is normal, one can estimate real and imaginary parts of eigenvalues of A in terms of eigenvalues of diagonal blocks A kk ; see also [8] where interesting analogue of min-max theorem for normal matrices was presented Theorem 22 Assume that A M n is normal and partitioned as in (21) Let σ(a) = {λ 1,λ 2,,λ n } Then, for any k {1,2,,q} and any α σ(a kk ), the following inequalities hold and min Re λ i Re α max Re λ i (24),,n,,n min Im λ i Im α max Im λ i (25),,n,,n Proof By the normality of A, there exist a unitary matrix U and a diagonal matrix D such that A = UDU with D = diag(λ 1,λ 2,,λ n ) Let y = U x, where x C n and x 2 = 1 Then y 2 = 1 and the Rayleigh quotient R(x,A) = x Ax can be written as R(x,A) = R(y,D) = n λ i y i 2 From this we obtain min Re λ i Re R(x,A) max Re λ i (26),,n,,n

Block Normal Matrices and Gershgorin-Type Discs 1063 and min Im λ i Im R(x,A) max Im λ i (27),,n,,n Assume now that u is an eigenvector of A kk such that u 2 = 1 and A kk u = αu Define x = [x 1 T,x 2 T,,x q T ] T, where x k C n k be such that x i = 0 for i k and x k = u Then R(x,A) = R(u,A kk ) = α From this and (26) (27), we get (24) (25) Notice that normality of A in Theorem 22 is crucial; see the following easy example Example 23 Let A = [ 1 2 2 3 ] Then A is not normal and the eigenvalues of A are λ 1,2 = 2 ± i 3 We see that (24) is not valid in this case Feingold and Varga proved the following Gershgorin-type theorem for partitioned matrices; see [3] and [11, pp 157 159] Theorem 24 Assume that A M n is partitioned as in (21) If λ σ(a) and λ / σ(a kk ) for all k, then there exists i {1,,q} such that 1 (λ I A ii ) 1 2 j i A ij 2 (28) However, it is difficult to compute this inclusion region in practice If all diagonal blocks A kk are normal, then the following theorem holds; see [5] Corollary 25 Assume that A M n is partitioned as in (21) Suppose that all diagonal blocks A kk of A are normal If λ is an eigenvalue of A, then there exists i {1,2,,q} such that min λ α α σ(a ii) A ij 2 (29) In [5], Meyer and Veselić obtained the following generalization of Theorem 12 Let Theorem 26 Assume that a normal matrix A M n is partitioned as in (21) Ŝ i (A) = A ji 2 2 (i = 1,,q) (210)

1064 J Kierzkowski and A Smoktunowicz Then each disc of the form z α Ŝi(A), where α is an eigenvalue of A ii, contains at least one eigenvalue of A Now we show that in (210) one can replace Ŝi(A) by smaller radii S i (A) Let Theorem 27 Assume that a normal matrix A M n is partitioned as in (21) S i (A) = [A T 1i,A T 2i,,A T i 1,i,A T i+1,i,,a T qi] 2 (i = 1,,q) (211) Then each disc of the form z α S i (A), where α is an eigenvalue of A ii, contains at least one eigenvalue of A Proof Let α be an eigenvalue of A ii Then there exists a vector u C ni such that A ii u = αu and u 2 = 1 Define x = [x 1 T,x 2 T,,x q T ] T, where x k C n k be such that x k = 0 for k i and x i = u Then Ax αx = A 1i u A 2i u A i 1,i u A ii u αu A i+1,i u A qi u = Taking norms, we get the desired inequality We have Thus, A 1i A 2i A i 1,i 0 A i+1,i A qi u Ax αx 2 [A T 1i,A 2i T,,A T i 1,i,0,A T i+1,i,,a T qi] T 2 u 2 Ax αx 2 S i (A) u 2 = S i (A) From this and Theorem 16, we conclude that This finishes the proof min λ i α Ax αx,,n 2 S i (A) Notice that by Ostrowski s theorem we have S i (A) Ŝi(A) for all i = 1,,q Now we give a block analogue of the inequality involving diagonal elements and eigenvalues of A; see [9] for unpartitioned case

Block Normal Matrices and Gershgorin-Type Discs 1065 Theorem 28 Assume that A M n is partitioned as in (21) and define R i (A) = A ij 2 2 (i = 1,,q) (212) Then each eigenvalue λ of A such that λ / σ(a ii ) for all i = 1,,q, satisfies (λ I ni A ii ) 1 2 2 R2 i (A) q q 1 (213) Proof Let λ be an eigenvalue of A, and suppose Ax = λx with x 2 = 1 Let x = [x 1 T,x 2 T,,x q T ] T, where x i C ni We have and thus, A ij x j = λx i, j=1 i = 1,,q, (λ I ni A ii )x i = A ij x j, i = 1,,q Since λ / σ(a ii ), all matrices λ I ni A ii are nonsingular From this, it follows that Taking norms, we obtain x i = (λ I ni A ii ) 1 A ij x j, i = 1,,q x i 2 (λ I ni A ii ) 1 2 A ij 2 x j 2 (214) By the Cauchy-Schwartz inequality, we have A ij 2 x j 2 2 so we can rewrite (214) as follows A ij 2 2 x i 2 2 (λ I ni A ii ) 1 2 2 R2 i (A) x j 2 2 = R2 i (A) x j 2 2, x j 2 2, i = 1,,q (215)

1066 J Kierzkowski and A Smoktunowicz For simplicity of notation, let b i stand for b i = x j 2 2 (216) Since 1 = x 2 2 = x 1 2 2 + + x q 2 2, the elements we have just defined satisfy b i = 1 x i 2 2, (217) and b i = (1 x i 2 2) = q x i 2 2 = q 1 (218) Thus, we can rewrite (215) as follows x i 2 2 (λ I ni A ii ) 1 2 2 R2 i (A) b i, i = 1,,q (219) Notice that by the assumptions of our theorem, all b i are positive From this and (217) (219), we get (λ I ni A ii ) 1 2 2 R2 i (A) = 1 b i b i = ( ) 1 1 b i 2 x i 2 = b i = 1 b i q (220) By an application of the harmonic-arithmetic mean inequality and from (218) we obtain 1 q 2 b q i b = q2 i q 1 This together with (220) gives the desired inequality (213) As a corollary we obtain the following generalization of Theorem 14 Theorem 29 If A M n satisfies the strong block square-sum criterion, ie, then A is nonsingular A 1 ii 2 2 R2 i (A) < 1, (221) Moreover, if A is Hermitian, satisfies the strong square-sum criterion and all diagonal blocks A ii are positive definite, then A is positive definite

Block Normal Matrices and Gershgorin-Type Discs 1067 Proof Let us first prove nonsingularity of A On the contrary, suppose that A is singular Then there is an eigenvalue λ = 0 of A From Theorem 28, taking λ = 0 in (213), we deduce that q/(q 1) < 1, which is impossible Now assume that A is Hermitian, satisfies the strong square-sum criterion and all diagonal blocks A ii are positive definite Since A is Hermitian, all its eigenvalues are real It is sufficient to prove that they are positive Conversely, suppose that there exists a negative eigenvalue ˆλ of A By assumption, all matrices A ii are Hermitian and positive definite, so ˆλ / σ(a ii ) Since ˆλ is negative, we conclude that By Theorem 28, we get (ˆλ I ni A ii ) 1 2 < A ii 1 2, i = 1,,q A 1 ii 2 2 R2 i (A) (ˆλ I ni A ii ) 1 2 2 R2 i (A) q q 1, but this contradicts our assumption (221) This finishes the proof Remark 210 Theorem 29 is an analogue of Varga s Theorem 62 for strictly block diagonally dominant matrices; see [11, p 157] It is clear that using A T instead of A in the above theorems a new set of the radii can be obtained It is easy to prove, by Ostrowski s theorem, that the strong block square-sum criterion (221) is sufficient condition for the convergence of the block Jacobi method for solving a linear system of equations Ax = b 3 Computational examples Numerical tests were given in MATLAB, version 650180913a (R13) with unit roundoff ǫ 22 10 16 in IEEE double precision The eigenvalues of A were plotted as crosses x Example 31 Let 611 0 0 407 0 0 49 0 899 113 192 0 0 8 0 113 899 196 0 0 8 A = 407 192 196 611 0 0 59 0 0 0 0 411 599 208 0 0 0 0 599 411 208 49 8 8 59 208 208 99 Then A is normal and σ(a) { 3747, 1224, 3029, 7295, 10100, 10120, 11390}

1068 J Kierzkowski and A Smoktunowicz Table 31 Results for the computed radii for the matrix A i 1 2 3 4 5 6 7 r i (A) 456 313 317 854 807 807 540 s i (A) 40993 22292 22638 49437 63408 63408 30420 800 600 600 400 400 200 200 0 0 200 200 400 600 400 800 500 0 500 1000 1500 600 400 200 0 200 400 600 800 1000 Picture 1: Gershgorin s discs G i (on the left) and the Euclidean discs E i (on the right) Notice that some eigenvalues of A are outside the set n E i Example 32 We generated random matrices of entries from the distribution N(0, 1) from Matlab s function randn We use the following Matlab s code to produce the matrix A(20 20), which is unitary in floating point arithmetic randn( state,0) n=20;a=orth(randn(n)+i*randn(n)); 4 3 1 2 05 1 0 0 1 05 2 3 1 4 5 4 3 2 1 0 1 2 3 4 5 15 1 05 0 05 1 15 Picture 2: Gershgorin s discs G i (on the left) and the Euclidean discs E i (on the right)

Block Normal Matrices and Gershgorin-Type Discs 1069 REFERENCES [1] L Cvetković, V Kostić, and R Varga A new Geršgorin-type eigenvalue inclusion set Electron Trans Numer Anal, 18:73 80, 2004 [2] E Deutsch Matricial norms Numer Math, 16:73 84, 1970 [3] DG Feingold and RS Varga Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem Pacific JMath, 12:1241 1250, 1962 [4] G Hämmerlin and Karl-Heinz Hoffmann Numerical Mathematics Springer-Verlag, 1991 [5] D Meyer and K Veselić On some new inclusion theorems for the eigenvalues of partitioned matrices Numer Math, 34:431 437, 1980 [6] AM Ostrowski On some metrical properties of operator matrices and matrices partitioned into blocks J Math Anal Appl, 2:161 209, 1961 [7] V Pták and J Zemánek Continuité Lipschitzienne du spectre comme fonction d un opérateur normal Commentationes Mathematicae Universitatis Carolinae, 17(3):507 512, 1976 [8] JF Queiró and AL Duarte Imbedding conditions for normal matrices Linear Algebra Appl, 430:1806 1811, 2009 [9] A Smoktunowicz An inequality involving diagonal elements and eigenvalues Solution 23-54 IMAGE: The Bulletin of the International Linear Algebra Society, 24:11 13, 2000 [10] A Smoktunowicz Block matrices and symmetric perturbations Linear Algebra Appl, 429:2628 2635, 2008 [11] RS Varga Geršgorin and His Circles Springer-Verlag, Berlin, 2004