IJMMS 28:5 (2001) 251–284 PII. S0161171201007062 http://ijmms.hindawi.com © Hindawi Publishing Corp. MATRIX SPLITTING PRINCIPLES ZBIGNIEW I. WOZNICKI´ (Received 15 March 2001) Abstract. The systematic analysis of convergence conditions, used in comparison theo- rems proven for different matrix splittings, is presented. The central idea of this analysis is the scheme of condition implications derived from the properties of regular splittings of a monotone matrix A = M1 −N1 = M2 −N2. An equivalence of some conditions as well as −1 ≥ −1 ≥ −1 ≥ −1 ≥ an autonomous character of the conditions M1 M2 0 and A N2 A N1 0 are pointed out. The secondary goal is to discuss some essential topics related with existing comparison theorems. 2000 Mathematics Subject Classification. 15A06, 15A18, 15A27, 65F10, 65F15, 65F50, 65N12. 1. Introduction. The main objective of this expository paper is to present the systematic analysis of convergence conditions derived from their implications for the regular splitting case and discussed in the subsequent sections. The secondary goal is to survey, compare and further develop properties of matrix splittings in order to present more clearly some aspects related with the results known in the literature. Consider the iterative solution of the linear equation system Ax = b, (1.1) where A ∈ Cn×n is a nonsingular matrix and x,b ∈ Cn. Traditionally, a large class of iterative methods for solving (1.1) can be formulated by means of the splitting A = M −N with M nonsingular, (1.2) and the approximate solution x(t+1) is generated, as follows Mx(t+1) = Nx(t) +b, t ≥ 0 (1.3) or equivalently x(t+1) = M−1Nx(t) +M−1b, t ≥ 0, (1.4) where the starting vector x(0) is given. The iterative method is convergent to the unique solution x = A−1b (1.5) for each x(0) if and only if (M−1N) < 1, which means that the used splitting of A = M −N is convergent. The convergence analysis of the above method is based on 252 ZBIGNIEW I. WOZNICKI´ the spectral radius of the iteration matrix (M−1N). For large values of t, the solution error decreases in magnitude approximately by a factor of (M−1N) at each iteration step; the smaller is (M−1N), the quicker is the convergence. Thus, the evaluation of an iterative method focuses on two issues: M should be chosen as an easily invertible matrix and (M−1N) should be as small as possible. General properties of a splitting of A = M − N (not necessary convergent), useful for proving comparison theorems, are given in the following theorem [20]. Theorem 1.1. Let A = M −N be a splitting of A ∈ Cn×n.IfA and M are nonsingular matrices, then M−1NA−1 = A−1NM−1, (1.6) the matrices M−1N and A−1N commute, and the matrices NM−1 and NA−1 also com- mute. Proof. From the definition of splitting of A, it follows that −1 −1 M−1 = (A+N)−1 = A−1 I +NA−1 = I +A−1N A−1 (1.7) or A−1 = M−1 +M−1NA−1 = M−1 +A−1NM−1 (1.8) which implies the equality (1.6) and hence, M−1NA−1N = A−1NM−1N or NM−1NA−1 = NA−1NM−1. (1.9) From the above theorem, the following results can be deduced. Corollary 1.2. Let A = M −N be a splitting of A ∈ Cn×n.IfA and M are nonsingular matrices, then by the commutative property, both matrices M−1N and A−1N (or NM−1 and NA−1) have the same eigenvectors. Lemma 1.3. Let A = M −N be a splitting of A ∈ Cn×n.IfA and M are nonsingular ma- −1 −1 trices, and λs and τs are eigenvalues of the matrices M N and A N, respectively, then τs λs = . (1.10) 1+τs Proof. By Corollary 1.2 we have −1 M Nxs = λs xs , (1.11) −1 A Nxs = τs xs . (1.12) The last equation can be written as −1 −1 −1 −1 A Nxs = I −M N M Nxs = τs xs (1.13) or equivalently −1 τs M Nxs = xs (1.14) 1+τs and the result (1.10) follows by the comparison of (1.11) with (1.14). MATRIX SPLITTING PRINCIPLES 253 From the above lemma, the following results can be concluded. Corollary 1.4. Let A = M − N be a splitting of A ∈ Cn×n, where A and M are −1 −1 nonsingular matrices, and λs and τs are eigenvalues of the matrices M N and A N, respectively. (a) If τs ∈ R, then the corresponding eigenvalue λs ∈ R, and conversely. (b) If τs ∈ C, then the corresponding eigenvalue λs ∈ C, and conversely. Remark 1.5. Let A = M − N be a splitting of A ∈ Cn×n, where A and M are non- −1 −1 singular matrices, and λs and τs are eigenvalues of the matrices M N and A N, re- −1 spectively. If the eigenvalue spectrum of A N is real, that is, τs ∈ R for all 1 ≤ s ≤ n, then the corresponding eigenvalues λs ∈ R, and conversely. Theorem 1.6. Let A = M − N be a splitting of A ∈ Cn×n, where A and M are non- −1 −1 singular matrices, and λs and τs are eigenvalues of the matrices M N and A N, respectively; and let the eigenvalue spectrum of A−1N be complex with the conjugate complex eigenvalues τs = as +bs i and τs+1 = as −bs i. The splitting is convergent if and only if 2 2 a 1+a +b2 b s s s + s < 1 (1.15) 2 2 2 2 1+as +bs 1+as +bs for all s = 1,3,5,...,n−1. Proof. By the relation (1.10), we have a +b i a 1+a +b2 b λ = s s = s s s + s i, s + + 2 2 2 2 1 as bs i 1+as +bs 1+as +bs (1.16) − + + 2 as bs i as 1 as bs bs λ +1 = = − i. s + − 2 2 2 2 1 as bs i 1+as +bs 1+as +bs Since for a convergent splitting −1 M N = max λs < 1 (1.17) s hence, inequality (1.15) follows immediately. Two particular cases of Theorem 1.6 are presented in the following lemmas. Lemma 1.7. Let A = M −N be a splitting of A ∈ Cn×n, where A and M are nonsingular −1 −1 matrices, and λs and τs are eigenvalues of the matrices M N and A N, respectively. If the eigenvalue spectrum of A−1N is real, then the splitting is convergent if and only if τs > −1/2 for all 1 ≤ s ≤ n. Proof. Since in this case λs ∈ R and τs ∈ R, then inequality (1.17) is satisfied when τs −1 <λs = < 1 (1.18) 1+τs which implies that τs > −1/2 for all 1 ≤ s ≤ n. 254 ZBIGNIEW I. WOZNICKI´ Lemma 1.8. Let A = M −N be a splitting of A ∈ Cn×n, where A and M are nonsingular −1 −1 matrices, and λs and τs are eigenvalues of the matrices M N and A N, respectively. If the eigenvalue spectrum of A−1N is complex with the purely imaginary eigenvalues τs = bs i and τs+1 =−bs i, then the splitting is convergent. Proof. It is evident that for this case inequality (1.15) reduces to the form 2 bs 2 < 1 (1.19) 1+bs which is satisfied for an arbitrary bs ∈ R. In the convergence analysis of iterative methods, the Perron-Frobenius theory of non- negative matrices plays an important role. This theory provides many theorems con- cerning the eigenvalues and eigenvectors of nonnegative matrices and its main results are recalled in the following two theorems [14]. Theorem 1.9. If A ≥ 0, then (1) A has a nonnegative real eigenvalue equal to its spectral radius, (2) To (A) > 0, there corresponds an eigenvector x ≥ 0, (3) (A) does not decrease when any entry of A is increased. Theorem 1.10. Let A ≥ 0 be an irreducible matrix. Then (1) A has a positive real eigenvalue equal to its spectral radius, (2) To (A) > 0, there corresponds an eigenvector x>0, (3) (A) increases when any entry of A increases, (4) (A) is a simple eigenvalue of A. In the case of nonnegative matrices the following theorem holds. Theorem 1.11. Let A = M − N be a splitting of A ∈ Rn×n, where A and M are nonsingular matrices. If both matrices M−1N and A−1N are nonnegative, then the splitting is convergent and A−1N M−1N = . (1.20) 1+ A−1N −1 Moreover, all complex eigenvalues τs of the matrix A N, if they exist, satisfy the in- equality 2 2 a 1+a +b2 b s s s + s ≤ 2 M−1N < 1 (1.21) 2 2 2 2 1+as +bs 1+as +bs −1 and all real eigenvalues τt of the matrix A N, if they exist, satisfy the inequality M−1N M−1N − ≤ τ ≤ . (1.22) 1+ M−1N t 1− M−1N Proof. From Theorem 1.9, we conclude that both nonnegative matrices M−1N −1 and A N have nonnegative dominant eigenvalues λ1 and τ1, respectively, equal to their spectral radii, and by Corollary 1.2 corresponding to the same nonnegative MATRIX SPLITTING PRINCIPLES 255 eigenvector x1 ≥ 0. Hence, by Lemma 1.3 the result (1.20) follows immediately. The same result (1.20) has been obtained by Varga [14] for regular splittings forty years ago.