Applied Mathematics Letters. Comparison theorems for a subclass of proper splittings of matrices

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Applied Mathematics Letters 25 (202) 2339 2343 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.

mparison theorems for a subclass of proper splittings of matrices Debasisha Mishra a,, K.C.

1 Applied Mathematics Letters 25 (202) Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: Comparison theorems for a subclass of proper splittings of matrices Debasisha Mishra a,, K.C. Sivakumar b a Institute of Mathematics and Applications, Bhubaneswar , India b Department of Mathematics, Indian Institute of Technology Madras, Chennai , India a r t i c l e i n f o a b s t r a c t Article history: Received 9 April 202 Received in revised form 30 June 202 Accepted 30 June 202 In this article, a convergence theorem and several comparison theorems are presented for a subclass of proper splittings of matrices introduced recently. 202 Elsevier Ltd. All rights reserved. Keywords: Moore Penrose inverse Comparison theorems Proper splitting Nonnegativity. Introduction and preliminaries Iterative methods for solving linear systems Ax = b, (.) where A is a real rectangular m n matrix and b is a real m-vector, are related to the decompositions of A of the form A = U V. A decomposition A = U V of A R m n is called a proper splitting [] if R(A) = R(U) and N(A) = N(U), where R(A) and N(A) stand for the range of A and the kernel of A. The asymptotic behavior of the iterative sequence x (i+) = U Ď Vx (i) + U Ď b, (.2) where U Ď is the Moore Penrose inverse [2] of U, is governed by the spectral radius of the iteration matrix U Ď V. (X Ď is the unique matrix which satisfies XX Ď X = X, X Ď XX Ď = X Ď, (XX Ď ) T = XX Ď and (X Ď X) T = X Ď X). The spectral radius of a real square matrix X is the maximum moduli of the eigenvalues of X, and is denoted by ρ(x). For a proper splitting A = U V, the iteration scheme (.2) converges to x = A Ď b for every initial vector x 0 if and only if ρ(u Ď V) < (see [, Corollary ]). When two decompositions or splittings of A are given, it is of interest to compare the spectral radii of the corresponding iteration matrices. The comparison of asymptotic rates of convergence of the iteration matrix induced by two splittings of a given matrix has been studied by many authors; for example, Csordas and Varga [3], Elsner [4], Song [5], Varga [6] and Woźnicki [7], to name a few. Here, splitting means a decomposition A = U V of a square matrix A where U is invertible. Woźnicki [8] has considered different types of splittings (such as regular, weak regular and weak nonnegative splittings of different types) of a given monotone matrix and has proved corresponding comparison theorems. Elsner [4] considered weak regular splittings and multisplittings, and proved comparison results. Song [5] studied a comparison theorem for nonnegative splittings and then applied it to study different basic iterative methods. In this article, we prove comparison results for different iterative schemes arising out of a new decomposition (called B Ď -splitting) introduced by Mishra and Sivakumar [9]. There, the authors have demonstrated the existence of a B Ď -splitting Corresponding author. address: kapamath@gmail.com (D. Mishra) /$ see front matter 202 Elsevier Ltd. All rights reserved. doi:0.06/j.aml

2 2340 D. Mishra, K.C. Sivakumar / Applied Mathematics Letters 25 (202) for a class of matrices, using a constructive procedure. Thus, the results that we have obtained in this article are very pertinent. However, applications of our results in the consideration of practical problems are devoted to a future study. The purpose of this article is to present a convergence theorem (Theorem 2.0) and two comparison theorems (Theorems 2.2 and 2.4) for B Ď -splittings. We say that a real matrix A is nonnegative, if it is entry-wise nonnegative, and we write this as A 0. The same notation and nomenclature are also used for vectors. If A and B are two real matrices, we write B A if B A 0. A decomposition A = U V of a real rectangular matrix A is called positive if U 0 and V 0. Before proving the main results, we recall certain definitions and results which will be used in Section 2. We begin with the definition of a B Ď -splitting. Definition. ([9, Definition 3.6]). A positive proper splitting A = U V of A R m n is called a B Ď -splitting if it satisfies the following conditions: (i) VU Ď 0, and (ii) Ax, Ux R m + + N(AT ) and x R(A T ) imply x 0. In the case of square nonsingular matrices, the above definition reduces to a B-splitting which we recall next. Definition.2 ([0, Definition ]). A positive splitting A = U V of A R n n is called a B-splitting if it satisfies the following conditions: (i) VU 0, and (ii) Ax, Ux 0 imply x 0. The next result guarantees the existence of B Ď -splittings for a certain class of matrices. Theorem.3 ([9, Theorem 3.0]). Let A R m n. Suppose that A Ď 0, R(A) int(r m + ) and AĎ A 0. Then A has a B Ď -splitting with ρ(vu Ď ) <. Now we recall the convergence theorem for semimonotone (meaning, A Ď 0) matrices using a B Ď -splitting of A. Theorem.4 ([9, Theorem 3.8]). Let A R m n. If A has a B Ď -splitting, then A Ď 0 if and only if ρ(vu Ď ) <. The next theorem is a part of the Perron Frobenius theorem. Theorem.5 ([6, Theorem 2.20]). Let A be a real square nonnegative matrix. Then we have the following. (i) A has a nonnegative real eigenvalue equal to its spectral radius. (ii) There exists a nonnegative eigenvector for its spectral radius. Another result which relates the spectral radius of two nonnegative matrices is given below. Theorem.6 ([6, Theorem 2.2]). Let A B 0. Then ρ(a) ρ(b). The result given next characterizes the reverse order law for the Moore Penrose inverse. Theorem.7 ([, Theorem ]). If A and B are real matrices such that AB is defined, then (AB) Ď = B Ď A Ď if and only if A Ď ABB T A T = BB T A T and BB Ď A T AB = A T AB. The following particular case will be useful in our discussion. Corollary.8. If A and B are real matrices such that AB is defined and B is invertible, then (AB) Ď = B A Ď if and only if A Ď ABB T A T = BB T A T. 2. Main results In this section we will prove a convergence theorem and various comparison theorems of B Ď -splittings. First, let us recall certain basic properties of proper splittings which form part of [, Theorem ]. Theorem 2.. Let A = U V be a proper splitting of A R m n. Then (a) A = U(I U Ď V); (b) I U Ď V is invertible; (c) A Ď = (I U Ď V) U Ď. It is evident that if A = U V is a proper splitting of A, then A T = U T V T is a proper splitting of A T. Next, we derive other properties of proper splittings that will be used in the rest of the paper.

3 D. Mishra, K.C. Sivakumar / Applied Mathematics Letters 25 (202) Theorem 2.2. Let A = U V be a proper splitting of A R m n. Then (a) AA Ď = UU Ď and A Ď A = U Ď U; (b) U = A(I + A Ď V) = (I + VA Ď )A; (c) I + A Ď V and I + VA Ď are invertible; (d) U Ď = (I + A Ď V) A Ď = A Ď (I + VA Ď ) ; (e) A Ď = (I + A Ď V)U Ď = U Ď (I + VA Ď ); (f) U Ď VA Ď = A Ď VU Ď ; (g) U Ď VA Ď V = A Ď VU Ď V ; (h) VU Ď VA Ď = VA Ď VU Ď. Proof. The proofs of (a), (g), (h) are trivial. (b) The fact that R(A) = R(U) implies that R(V) R(A). Hence V = AA Ď V. Also, R(A T ) = R(U T ) implies R(V T ) R(A T ). It then follows, as above, that V T = A T (A Ď ) T V T = (A Ď A) T V T which yields V = VA Ď A. So, U = A + V = A(I + A Ď V) = (I + VA Ď )A. (c) Now, we show that I + A Ď V is invertible. Suppose that (I + A Ď V)x = 0. Then x = A Ď Vx R(A Ď ) = R(A T ) = R(U T ). Hence x = U Ď Ux = A Ď Ax. So x = A Ď Vx = A Ď (U A)x = A Ď Ux A Ď Ax = A Ď Ux x. Thus A Ď Ux = 0, so that Ux N(A Ď ) = N(A T ) = N(U T ) = N(U Ď ). Therefore, x = U Ď Ux = 0. Hence I + A Ď V is invertible. By applying the first part to the decomposition A T = U T V T one gets that I + (A T ) Ď V T = (I + VA Ď ) T is invertible. Thus I + VA Ď is invertible. (d) Since R(A Ď ) = R(A T ) = R(U T ), we have R((I +A Ď V)(I +A Ď V) T A T ) = R((I +A Ď V)U T ) = R(U T +A Ď VU T ) R(U T ) = R(A T ). Thus A Ď A(I + A Ď V)(I + A Ď V) T A T = (I + A Ď V)(I + A Ď V) T A T. Then by Corollary.8, U Ď = (I + A Ď V) A Ď. Similarly, U Ď = A Ď (I + VA Ď ). (e) A Ď = U Ď (I + VA Ď ) and A Ď = (I + A Ď V)U Ď follow from (d). (f) We have A Ď = U Ď (I + VA Ď ) = U Ď + U Ď VA Ď and A Ď = (I + A Ď V)U Ď = U Ď + A Ď VU Ď. Hence U Ď VA Ď = A Ď VU Ď. Corollary 2.3. Let A = U V be a proper splitting of A R m n. Then the matrices U Ď V and A Ď V (or VU Ď and VA Ď ) have the same eigenvectors. Theorem 2.2 is a generalization of the next result which holds for nonsingular matrices. Corollary 2.4 ([7, Lemma.]). Let A = U V be a splitting of A R n n. Suppose that A and U are nonsingular matrices. Then (a) U VA = A VU ; (b) U VA V = A VU V ; (c) VU VA = VA VU. Corollary 2.5 ([7, Corollary.]). Let A = U V be a splitting of A R n n. If A and U are nonsingular matrices, then the matrices U V and A V (or VU and VA ) have the same eigenvectors. Lemma 2.6. Let A = U V be a proper splitting of A R m n. Let µ i, i s and λ j, j s be the eigenvalues of the matrices U Ď V (VU Ď ) and A Ď V (VA Ď ) respectively. Then for every j, we have + λ j 0. Also, for every i, there exists j such that µ i = λ j and for every j, there exists i such that +λ j λ j = µ i µ i. Proof. Let x be an eigenvector corresponding to the eigenvalue µ i of the matrix U Ď V. Then from Corollary 2.3, we have µ i x = U Ď Vx = (I + A Ď V) A Ď Vx = λ j +λ j x for some eigenvalue λ j of A Ď V. The second part follows similarly. Let A = U V = U 2 V 2 be two different decompositions of A R m n. Define S := {j N : (A Ď V 2 ) j A Ď (A Ď V ) j A Ď }. Lemma 2.7. Suppose that A Ď 0, V 0 and V 2 0. Then S is closed under addition. Proof. Let j, k S and j, k. Then (A Ď V 2 ) j A Ď (A Ď V ) j A Ď and (A Ď V 2 ) k A Ď (A Ď V ) k A Ď. Post-multiplying the first inequality by V 2 (A Ď V 2 ) k A Ď, we get (A Ď V 2 ) j+k A Ď (A Ď V ) j (A Ď V 2 ) k A Ď. Pre-multiplying the second inequality by (A Ď V ) j, we get (A Ď V ) j (A Ď V 2 ) k A Ď (A Ď V ) j+k A Ď. Combining, we get (A Ď V 2 ) j+k A Ď (A Ď V ) j+k A Ď so that j + k S. The next theorem provides sufficient conditions under which S is shown to be nonempty. Theorem 2.8. Let A = U V = U 2 V 2 be two proper splittings of A R m n. (a) Let U Ď 0, U Ď 2 0 and V 2 V. Then U Ď U Ď 2. (b) Let A Ď 0, V 2 0 and V 0. If U Ď U Ď 2 then (AĎ V 2 ) j A Ď (A Ď V ) j A Ď, for each positive integer j.

4 2342 D. Mishra, K.C. Sivakumar / Applied Mathematics Letters 25 (202) Proof. (a) Since A = U V = U 2 V 2 are two proper splittings of A, we have R(U ) = R(U 2 ) = R(A), R(U T ) = R(U T 2 ) = R(A T ), U 2 = U U Ď U 2 and U = U U Ď 2 U 2. Further, U U Ď = U 2U Ď 2. Also, U 2 U = (A + V 2 ) (A + V ) = V 2 V 0. So, 0 U 2 U = U 2 U Ď 2 U 2 U U Ď U = U U Ď U 2 U U Ď 2 U 2 = U (U Ď U Ď 2 )U 2. Pre-multiplying by U Ď, we get 0 U Ď U (U Ď U Ď 2 )U 2 = (U Ď U Ď 2 )U 2. Post-multiplying by U Ď 2, we get 0 (U Ď U Ď 2 )U 2U Ď 2 = U Ď U Ď. 2 Therefore U Ď U Ď. 2 (b) Let A Ď 0, V 0 and V 2 0. If U Ď U Ď 2, then (from Theorem 2.2 (d)) (I + AĎ V ) A Ď A Ď (I + V 2 A Ď ). Hence A Ď (I + V 2 A Ď ) (I + A Ď V )A Ď, i.e., A Ď + A Ď V 2 A Ď A Ď + A Ď V A Ď so that A Ď V 2 A Ď A Ď V A Ď. Thus S. By Lemma 2.7, it follows that (A Ď V 2 ) j A Ď (A Ď V ) j A Ď, for each positive integer j. A splitting A = U V of A R n n is called a regular splitting if U 0 and V 0. We obtain the following result for regular splittings of a monotone matrix as a corollary to the theorem above. Corollary 2.9 ([3, Proposition ]). Let A = U V = U 2 V 2 be regular splittings of A, where A 0. Then, (a) V 2 V implies U U 2 ; (b) U U 2 implies (A V 2 ) j A (A V ) j A, for each positive integer j. Now, we present a convergence theorem for a B Ď -splitting. Theorem 2.0. Let A = U V be a B Ď -splitting of A R m n. If A Ď 0, then we have the following. (a) A Ď U Ď. (b) ρ(va Ď ) ρ(vu Ď ). (c) ρ(vu Ď ) = ρ(u Ď V) = ρ(aď V) +ρ(a Ď V) <. Proof. Let A = U V be a B Ď -splitting. Then R(A) = R(U), N(A) = N(U), U 0, V 0 and VU Ď 0. Also, Ax, Ux R m + + N(AT ) and x R(A T ) imply x 0. (a) We have A Ď U Ď = (I + A Ď V)U Ď U Ď = A Ď VU Ď 0, where the last inequality is valid due to A Ď 0. So, A Ď U Ď. (b) Since V 0, so pre-multiplying the inequality A Ď U Ď by V, we have VA Ď VU Ď 0. By Theorem.6, it then follows that ρ(va Ď ) ρ(vu Ď ). λ +λ (c) Let λ be any eigenvalue of VA Ď. Let f (λ) =, λ 0. Then f is a strictly increasing function. There exists an eigenvalue of VU Ď, say µ, such that µ = λ. So µ attains its maximum when λ is maximum. But λ is maximum when +λ λ = ρ(a Ď V) = ρ(va Ď ). (Note that by Theorem.5, ρ(va Ď ) is an eigenvalue of VA Ď.) As a result, the maximum value of µ is ρ(vu Ď ) = ρ(u Ď V). Hence, ρ(u Ď V) = ρ(aď V) +ρ(a Ď V) <. Note that (c) of Theorem 2.0 is already known for outer generalized inverses. (See, for instance, [2, Theorem 3.5.9]) When A is nonsingular, we obtain the following new result, as a corollary to the above theorem for a B-splitting. Corollary 2.. Let A = U V be a B-splitting of A R n n. If A 0, then we have the following. (a) A U. (b) ρ(va ) ρ(vu ). (c) ρ(vu ) = ρ(u V) = ρ(a V) +ρ(a V) <. Next, we present comparison theorems for B Ď -splittings. Theorem 2.2. Let A = U V = U 2 V 2 be B Ď -splittings of A R m n. If V V 2 and A Ď 0, then ρ(u Ď V ) ρ(u Ď Proof. We have ρ(u Ď i V i) < for i =, 2 by Theorem 2.0. Observe that A Ď V 2 A Ď V 0. Then ρ(a Ď V 2 ) ρ(a Ď V ). As in ρ(a Theorem 2.0, it follows that Ď V ) +ρ(a Ď V ) ρ(aď V 2 ) +ρ(a Ď V 2 ). Corollary 2.3. Let A = U V = U 2 V 2 be B-splittings of A R n n. If V V 2 and A 0, then ρ(u V ) ρ(u Theorem 2.4. Let A = U V = U 2 V 2 be B Ď -splittings of A R m n, where A Ď 0. If U Ď U Ď 2, then ρ(u Ď V ) ρ(u Ď Proof. By Theorem 2.0, we have ρ(u Ď i V i) < for i =, 2. Also V i A Ď 0 for i =, 2. Clearly, it suffices to show that ρ(a Ď V 2 ) ρ(a Ď V ). We have (I + A Ď V ) A Ď = U Ď U Ď = 2 AĎ (I + V 2 A Ď ). Thus A Ď (I + V 2 A Ď ) (I + A Ď V )A Ď i.e., A Ď V 2 A Ď A Ď V A Ď. Post-multiplying by V 2, we have (A Ď V 2 ) 2 A Ď V A Ď V 2. On the other hand, post-multiplying by V, we get A Ď V 2 A Ď V (A Ď V ) 2. Therefore ρ((a Ď V 2 ) 2 ) ρ(a Ď V A Ď V 2 ) = ρ(a Ď V 2 A Ď V ) ρ((a Ď V ) 2 ). Hence ρ(a Ď V 2 ) ρ(a Ď V ).

5 D. Mishra, K.C. Sivakumar / Applied Mathematics Letters 25 (202) Corollary 2.5. Let A = U V = U 2 V 2 be B-splittings of A R n n, where A 0. If U U 2, then ρ(u V ) ρ(u Finally, we impose still weaker hypotheses yielding generalizations of Theorems 2.2 and 2.4, using Theorem 2.8. The proof of the first one is trivial while the other one is a particular case of the first. Corollary 2.6. Let A = U V = U 2 V 2 be B Ď -splittings of A R m n, where A Ď 0. If there exists a positive integer j such that (A Ď V 2 ) j A Ď (A Ď V ) j A Ď, then ρ(u Ď V ) ρ(u Ď Corollary 2.7. Let A = U V = U 2 V 2 be B-splittings of A R n n, where A 0. If there exists a positive integer j such that (A V 2 ) j A (A V ) j A, then ρ(u V ) ρ(u Acknowledgments We thank the referees for their helpful comments and suggestions that have led to a much improved presentation. References [] A. Berman, R.J. Plemmons, Cones and iterative methods for best square least squares solutions of linear systems, SIAM J. Numer. Anal. (974) [2] A. Ben-Israel, T.N.E. Greville, Generalized Inverses. Theory and Applications, Springer-Verlag, New York, [3] G. Csordas, R.S. Varga, Comparisons of regular splittings of matrices, Numer. Math. 44 (984) [4] L. Elsner, Comparisons of weak regular splittings and multi splitting methods, Numer. Math 56 (989) [5] Y. Song, Comparisons of nonnegative splittings of matrices, Linear Algebra Appl (99) [6] R.S. Varga, Matrix Iterative Analysis, Springer-Verlag, Berlin, [7] Z.I. Woźnicki, Matrix splitting principles, Novi Sad J. Math. 28 (998) [8] Z.I. Woźnicki, Nonnegative splitting theory, Japan J. Indust. Appl. Math. (994) [9] D. Mishra, K.C. Sivakumar, On splittings of matrices and nonnegative generalized inverses, Oper. Matrices 6 () (202) [0] J.E. Peris, A new characterization of inverse-positive matrices, Linear Algebra Appl (99) [] T.N.E. Greville, Note on generalized inverse of a matrix product, SIAM Rev. 8 (966) [2] D.S. Djordjević, V. Rakočević, Lectures on generalized inverses, Faculty of Sciences and Mathematics, Nis, 2008.

Weak Monotonicity of Interval Matrices

Electronic Journal of Linear Algebra Volume 25 Volume 25 (2012) Article 9 2012 Weak Monotonicity of Interval Matrices Agarwal N. Sushama K. Premakumari K. C. Sivakumar kcskumar@iitm.ac.in Follow this and