An estimation of the spectral radius of a product of block matrices
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1 Linear Algebra and its Applications 379 (2004) wwwelseviercom/locate/laa An estimation of the spectral radius of a product of block matrices Mei-Qin Chen a Xiezhang Li b a Department of Mathematics and Computer Science The Citadel Charleston SC USA b Department of Mathematical Sciences Georgia Southern University Statesboro GA USA Received 29 October 2002; accepted 14 April 2003 Submitted by F Uhlig Abstract Let C(r) =[C ij ] r = 1 2Rbeblockm m matrices where C ij (r) are nonnegative N i N j matrices for i j = 1 2m Let be a consistent matrix norm Denote for each r by B(r) =[ C ij (r) ] an m m matrix The relation of the spectral radii ρ( R r=1 C(r)) and ρ( R r=1 B(r)) is studied in this paper It is shown with two proofs that R R ρ C(r) ρ B(r) r=1 r=1 As shown in one of the proofs ρ( R r=1 B(r)) can be reduced so that it gives a better estimation of ρ( R r=1 C(r)) 2003 Elsevier Inc All rights reserved Keywords: Nonnegative matrices; Spectral radius; Block matrices 1 Introduction Let {N 1 N m } be a set of positive integers and N be a positive integer Let C ij be real N i N j matrices for i j = 1mThen the matrix This research was partially supported by a grant from the Citadel Foundation Corresponding author addresses: chenm@citadeledu (M-Q Chen) xli@gsucsgasouedu (X Li) /$ - see front matter 2003 Elsevier Inc All rights reserved doi:101016/s (03)
2 268 M-Q Chen X Li / Linear Algebra and its Applications 379 (2004) C 11 C 1m C =[C ij ]= (11) C m1 C mm is called a block m m matrix and a square block m m matrix when C ij are square matrices of order N Each C ij is referred to as the (i j) block of C Let C 11 (r) C 1m (r) C(r) =[C ij (r)] = r = 1R (12) C m1 (r) C mm (r) be R block m m matrices where C ij (r) are real N i N j matrices If a matrix P =[p kl ] then P is the nonnegative matrix whose entries are p kl Inmany applications the spectral radius ρ( R r=1 C(r)) needs to be computed or estimated For example [1] in the design of two-dimensional (2-D) digital filters with periodic coefficients in signal processing one of the sufficient conditions for zeroinput stability of a 2-D digital filter with periodic coefficients is ρ( R r=1 C(r) ) <1 where C ij (r) are functions of parameters of the filter and R is the greatest common divisor of two periods The computational complexity of ρ( R r=1 C(r) ) increases as the size of C ij (r) or R increases It is important to find an estimate of ρ( R r=1 C(r) ) with less computations In this paper we investigate estimations of ρ( C ) and ρ( R r=1 C(r) ) Throughout the paper we call a consistent matrix norm if for all rectangular matrices A and B it satisfies the following four axioms: (1) A 0 and A =0 if and only if A = 0; (2) ca = c A for any scalar c; (3) A + B A + B where A and B are in the same size; and (4) AB A B provided that AB is defined Since ρ(c) ρ( C ) without loss of generality we assume that C and C(r) for r = 1Rare nonnegative from now on Denote by m m matrices B =[ C ij ] and B(r) =[ C ij (r) ] for r = 1Rwhere is a consistent matrix norm Inequalities ρ(c) ρ(b) and ρ( R r=1 C(r)) ρ( R r=1 B(r)) are derived in Section 2 Clearly the computational complexities of ρ(b) and ρ( R r=1 B(r)) are much less than the ones of ρ(c) and ρ( R r=1 C(r)) respectively So ρ(b) and ρ( R r=1 B(r)) can be used to estimate ρ(c) and ρ( R r=1 C(r)) respectively In Section 3 an alternative proof for the inequality ρ(c) ρ(b) with 1 and is given The proof suggests a way to reduce ρ(b) and ρ( R r=1 B(r)) to improve their estimations to ρ(c) and ρ( R r=1 C(r)) respectively Numerical examples are discussed to illustrate the findings
3 M-Q Chen X Li / Linear Algebra and its Applications 379 (2004) Estimations of ρ(c) and ρ( R r=1 C(r)) Theorem 21 Let C =[C ij ] be a block m m matrix where C ij are nonnegative N i N j matrices Let B =[ C ij ] a nonnegative m m matrix where is a consistent matrix norm Then ρ(c) ρ(b) (21) Proof First we consider the case where C is nonnegative and irreducible and each C ij is a square matrix of order N By the Perron Frobenius Theorem [3] ρ(c) is an eigenvalue corresponding to a positive vector x in R Nm ie Cx = ρ(c)x Let x =[x1 TxT m ]T where x i R N and z =[z 1 z m ] T R m wherez i = x i for i = 1mThen for 1 i m C ij x j = ρ(c)x i which implies ρ(c) z i = ρ(c) x i C ij x j C ij x j = C ij z j Since the inequality ρ(c)z i m C ij z j holds for all i = 1mwe have ρ(c) z Bz Since B is nonnegative and z>0[2] ρ(c) ρ(b) Secondly we show that the inequality (21) holds for all nonnegative matrices C where C ij s are square Let C ij =[c (ij) kl ] for i j = 1mFor each ɛ>0 define C ij (ɛ) := [ c (ij) kl + ɛ ] for all i j = 1m and C(ɛ) := [C ij (ɛ)] and B(ɛ) := [ C ij (ɛ) ] m ij Since C ij (ɛ) s are positive they are irreducible for all i j and therefore ρ(c(ɛ)) ρ(b(ɛ)) By the continuity of ρ we have ρ(c) = lim ɛ 0 ρ(c(ɛ)) lim ɛ 0 ρ(b(ɛ)) = ρ(b)
4 270 M-Q Chen X Li / Linear Algebra and its Applications 379 (2004) Finally we show that the inequality (21) holds for all nonnegative matrices C where C ij s are N i N j matrices Let N = max 1 i m {N i } Define C =[C ij ] a block m m matrix where each C ij is a square matrix of order N and Cij 0 C ij = 0 0 Clearly C ij = C ij for all i j Since there exists a permutation matrix P such that P CP T C 0 = 0 0 ρ(c) = ρ(c) Then B := [ C ij ] = B Since C is a nonnegative square block matrix we have ρ(c) ρ(b) = ρ(b) Therefore ρ(c) ρ(b) The inequality (21) also holds for a product of block matrices C(r) as defined in (12) For simplicity a product of two matrices is considered in the following theorem Theorem 22 Let C(r) =[C ij (r)] be block m m matrices for r = 1 2 where C ij (r) are nonnegative N i N j matrices for i j = 1m Define B(r) = [ C ij (r) ] for r = 1 2 and is a consistent matrix norm Then ρ(c(2)c(1)) ρ(b(2)b(1)) (22) Proof It suffices to show that the norm of the (i j) block of C(2)C(1) is less than or equal to the (i j) element of B(2)B(1) Observe that the (i j) block of C(2)C(1) is m C ik (2)C kj (1) Then C ik (2)C kj (1) m C ik (2)C kj (1) C ik (2) C kj (1) = the (i j) element of B(2)B(1) By Theorem 21 the proof of (22) is completed Theorem 21 with induced matrix norms was first presented in [4] We extend the result to any consistent matrix norm and give a completely different proof In addition we generalize the result to a product of block matrices In practice Frobenius norm F 1-norm 1 and -norm are commonly used For many applications the estimate ρ(b) using F which is not an induced matrix norm is better than the ones using 1 and For example consider
5 M-Q Chen X Li / Linear Algebra and its Applications 379 (2004) C11 (r) C C(r) = 12 (r) C 21 (r) C 22 (r) r = 1 2 (23) where C 11 (1) = C (1) = C 21 (1) = C (1) = C 11 (2) = C (2) = C 21 (2) = C (2) = Let B p (r) =[ C ij (r) p ] for i j r = 1 2 where p = or F Then we have ρ(b (2)B (1)) = and ρ(b F (2)B F (1)) = (24) 3 Improving the estimation of ρ(c) and ρ( R r=1 C(r)) Because of the inequality (22) ρ( R r=1 C(r)) can be estimated by ρ( R r=1 B(r)) For some applications ρ( R r=1 B(r)) may be much larger than ρ( R r=1 C(r)) In this section we give an alternative proof for the inequality (21) with 1 or The proof describes how much ρ(c) is enlarged and suggests a way to reduce ρ(b) It is known [2] that if row sums of a nonnegative square matrix P =[p ij ] of order Nare constant ie N p ij = c for i = 1N then ρ(p) = P = c This result can be generalized to square block matrices as given in Lemma 31 Lemma 31 Let A =[A ij ] be block m m matrix where A ij are nonnegative square matrices of order N If the row sums of A ij are b ij for i j = 1mand B =[b ij ] an m m matrix is irreducible then ρ(a) = ρ(b) (31) Proof Let e N =[11] T R N By assumption A ij e N = b ij e N for all i j = 1m For v =[v 1 v 2 v m ] T R m we have
6 272 M-Q Chen X Li / Linear Algebra and its Applications 379 (2004) m v j A 1j e m N v j b 1j e N A(v e N ) = = m v j A mj e m N v j b mj e N ( m ) v j b 1j en = ( = (Bv) e N m ) v j b mj en If (λ v) is an eigenpair of B ie Bv = λv then A(v e N ) = (Bv) e N = (λv) e N = λ(v e N ) (32) Therefore (λ v e N ) is an eigenpair of A Since B is nonnegative and irreducible by the Perron Frobenius Theorem [3] there exists v>0suchthat Bv = ρ(b)v It follows from (32) that A(v e N ) = ρ(b)(v e N ) It is also known [2] that if a nonnegative and irreducible matrix has a positive eigenvector then the corresponding eigenvalue of this positive eigenvector is the spectral radius of the matrix The vector v>0implies v e N > 0 and therefore ρ(b) is ρ(a) It completes the proof The equality (31) also holds for a product of block matrices For simplicity a product of two matrices is considered in the following lemma Lemma 32 LetA(r) =[A ij (r)] be block m m matrices for r = 1 2 where A ij (r) are nonnegative square matrices of order N for i j = 1m Suppose that row sums of A ij (r) are b ij (r) for all i j = 1mand r = 1 2 and B(r) =[b ij (r)] m m matrices and B(2)B(1) is irreducible Then ρ(a(2)a(1)) = ρ(b(2)b(1)) Proof It suffices to show the row sums of the (i j) block of A(2)A(1) are equal to the (i j) element of B(2)B(1) for all i j = 1 m Observe that the (i j) block of A(2)A(1) is m A ik (2)A kj (1) Since ( m ) A ik (2)A kj (1) e N = A ik (2)(A kj (1)e N ) = A ik (2)(b kj (1)e N ) = (A ik (2)e N )b kj (1) = (b ik (2)e N )b kj (1)
7 M-Q Chen X Li / Linear Algebra and its Applications 379 (2004) ( m ) = b ik (2)b kj (1) e N the row sums of the (i j) block of A(2)A(1) are m b ik (2)b kj (1) which is the (i j) element of B(2)B(1) Using Lemma 31 we can prove Theorem 21 by defining a block m m matrix A =[A ij ] where A ij are nonnegative N i N j matrices such that C A and A ij = b ij for i j = 1mDetails are given in the following theorem Theorem 31 Let C =[C ij ] be a block m m matrix where C ij are nonnegative N i N j matrices for i j = 1mand let B =[ C ij ] an m m matrix where is either 1 or Then ρ(c) ρ(b) (33) Proof In an analogous way as the proof of Theorem 21 it suffices to show the inequality (33) holds if C ij s are square matrices of order N and B is irreducible First we consider the case where = Let C ij =[c (ij) kl ] be square matrices of order N Define A =[A ij ] ablockm m matrix and A ij = D ij C ij where D ij = diag { d (ij) 1 d (ij) } N and Since d (ij) k = N t=1 C ij Nt=1 c (ij) 1 otherwise if N t=1 c (ij) /= 0 c (ij) C ij for k = 1N for k = 1N C ij A ij for all i j = 1mHence C A and ρ(c) ρ(a) Observe that row sums of A ij are C ij for all i j = 1m Since B = [ C ij ] is irreducible it follows directly from Lemma 31 that ρ(a) = ρ(b) Therefore ρ(c) ρ(b) The proof for the case where = 1 is the same with using the transposes of A B and C based on the facts that for any square matrix M ρ(m) = ρ(m T ) and M T 1 = M
8 274 M-Q Chen X Li / Linear Algebra and its Applications 379 (2004) The inequality (33) also holds for a product of block matrices The proof is directly from Lemma 32 and Theorem 31 We state below the result with a product of two matrices without giving a proof Theorem 32 Let C(r) =[C ij (r)] be block m m matrices for r = 1 2 where C ij (r) are nonnegative N i N j matrices for all i j = 1m and let B(r) = [ C ij (r) ] m m matrices for r = 1 2 where is either 1 or Then ρ(c(2)c(1)) ρ(b(2)b(1)) The proof of Theorem 31 describes how the matrix C is enlarged to the matrix A Notice the following: A ij = C ij for i j = 1mwhere the same matrix norm is used in defining A For i j = 1mA ij = D ij C ij max 1 k N {d (ij) k }C ij If all ratios d (ij) are close to 1 then ρ(c) ρ(a) = ρ(b) So if the product of the ratios R m k N r=1 i ( d (ij) k (r) ) (34) can be reduced then the value of ρ( R r=1 B(r)) can also be reduced To reduce the value of ρ( R r=1 B(r)) we may want to find a permutation matrix P such that ρ(b(r)b(r 1) B(2)B(1)) < ρ(b(r) B(1)) where B(R) =[ C ij (R) ] C(R) = P C(R) and B(1) =[ C ij (1) ] C(1) = C(1)P T One of the ways to form P is to switch rows of C(R) or switch columns of C(1) so that ( ) m N m ( ) Cij (R) N Cij (1) Nt=1 i c (ij) (R) Nt=1 i c (ij) (1) is reduced We further explain this possible reduction through the following two examples Example 31 Consider C(1) and C(2) given in (23) Asgivenin(24) the estimates of ρ(c(2)c(1)) using and F are ρ(b (2)B (1)) = and ρ(b F (2)B F (1)) = respectively To reduce ρ(b (2)B (1)) we let P be the permutation matrix that switches the second and the third row of C(2) and let
9 M-Q Chen X Li / Linear Algebra and its Applications 379 (2004) C(2) = PC(2) C(1) = C(1)P B (2) =[ C ij (2) ] B (1) =[ C ij (1) ] (35) Then the product of ratios defined in (34) is reduced from to 1792 and ρ(b (2)B (1)) = Example 32 We change C(1) in Example 31 as follows: C 11 (1) = C (1) = C 21 (1) = C (1) = Then ρ(c(2)c(1)) = and its estimates using and F are ρ(b (2)B (1)) = and ρ(b F (2)B F (1)) = respectively Let P again be the permutation matrix that switches the second and third row of C(2) and define B (1) and B (2) as in (35) Then the product of ratios defined in (34) is reduced from to 1008 and ρ(b (2)B (1)) = Acknowledgement The authors would like to thank the referee who suggested a number of improvements to the paper References [1] T Bose M-Q Chen KS Joo G-F Xu Stability of two-dimensional discrete systems with periodic coefficients IEEE Trans Circuits and Systems Part II: Analog and Digital Signal Processing 45 (7) (1998) [2] R Horn C Johnson Matrix Analysis Cambridge University Press Cambridge 1985 [3] R Varga Matrix Iterative Analysis Springer Berlin 2000 [4] Z-Y You Block estimation of the spectral radius of a matrix Numerical Math J Chinese Univ 1 (1) (1979)
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