DOCSLIB.ORG

Characterization and Properties of .R; S /-Commutative Matrices

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Characterization and Properties of .R; S /-Commutative Matrices Characterization and properties of .R;S/-commutative matrices William F. Trench Trinity University, San Antonio, Texas 78212-7200, USA Mailing address: 659 Hopkinton Road, Hopkinton, NH 03229 USA NOTICE: this is the author’s version of a work that was accepted for publication in Linear Algebra and Its Applications. Changes resulting from the publishing process, such as peer review, editing,corrections, structural formatting,and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. Please cite this article in press as: W.F. Trench, Characterization and properties of .R;S /-commutative matrices, Linear Algebra Appl. (2012), doi:10.1016/j.laa.2012.02.003 Abstract 1 Cmm Let R D P diag. 0Im0 ; 1Im1 ;:::; k1Imk1 /P 2 and S D 1 Cnn Q diag. .0/In0 ; .1/In1 ;:::; .k1/Ink1 /Q 2 , where m0 C m1 C C mk1 D m, n0 C n1 CC nk1 D n, 0, 1, ..., k1 are distinct mn complex numbers, and W Zk ! Zk D f0;1;:::;k 1g. We say that A 2 C is .R;S /-commutative if RA D AS . We characterize the class of .R;S /- commutative matrrices and extend results obtained previously for the case where 2i`=k ` D e and .`/ D ˛` C .mod k/, 0 Ä ` Ä k 1, with ˛, 2 Zk. Our results are independent of 0, 1,..., k1, so long as they are distinct; i.e., if RA D AS for some choice of 0, 1,..., k1 (all distinct), then RA D AS for arbitrary of 0, 1,..., k1. MSC: 15A09; 15A15; 15A18; 15A99 Keywords: Commute; Eigenvalue problem; Least Squares problem; Moore–Penrose Inverse; .R;S /-commutative; Singular value decomposition e-mail:[email protected] 1 2 William F. Trench 1 Introduction n1 nn A matrix A Œars C is said to be centrosymmetric if D r;sD0 2 anr1;ns1 ars ; 0 r; s n 1; D Ä Ä or centro-skewsymmetric if anr1;ns1 ars; 0 r; s n 1: D Ä Ä The study of such matrices is facilitated by the observation that A is centrosymmetric (centro-skewsymmetric) if and only if JA AJ (JA AJ ), where J is the flip matrix, with ones on the secondary diagonalD and zeroesD elsew here. Several authors [2, 3, 4, 5, 8, 10, 13, 25] used thisobservationto showthat centrosymmetric and centro- skewsymmetric matrices can be written as A P CP 1, where P diagonalizes J and C has a useful block structure. We will discussD this further in Example 3. Followingthis idea, other authors [6, 11, 12, 14, 24] considered matrices satisfying RA AR or RA AR, where R is a nontrivial involution; i.e., R R1 I . We continuedD this lineD of investigaton in [15, 16, 17, 19], and extendedD it in [18,¤˙ 20], defining A Cmn to be .R; S/-symmetric (.R; S/-skew symmetric) if RA AS (RA AS2), where R Cmm and S Cnn are nontrivialinvolutions. We showedD that aD matrix A with either2 of these properties2 can be written as A P CQ1, where P and Q diagonalize R and S respectively and C has a useful blockD form. Chen [7] and Fasino [9] studied matrices A Cnn such that RAR A, where R is a unitary matrix that satisfies Rk I 2for some k n and e2i=kD . In [21] we studied matrices A Cmn such thatDRA AS, whereÄ D 2 D k1 1 R P diag Im ; Im ;:::; Im P ; (1) D 0 1 k1 Á k1 1 S Q diag In ; In ;:::; In Q ; (2) D 0 1 k1 Á m0 m1 mk1 m; n0 n1 nk1 n; (3) C CC D C CC D and ˛; Zk 0;1;:::;k 1 : 2 Df g Finally, motivated by a problem concerning unilevel block circulants [22], in [23] we mn ˛ considered matrices A C such that RA AS , with ˛; Zk. We called such matrices .R;S;˛;/2 -symmetric, and showedD that A has this property2 if and only if k1 ˛`C. mod k/n` A P˛`C. k/F`Q` with F` C ; 0 ` k 1; (4) D mod 2 Ä Ä D X` 0 b which has useful computational and theoretical applications. (P0, ..., Pk1 and Q0, ..., Qk1 are defined in Section 2, specifically, (7)–(10).) The class of .R;S;˛;/- symmetric matrices includes, for example, centrosymmetric, skew-centrosymmetric,b b .R;S /-commutative matrices 3 R-symmetric, R-skew symmetric, .R; S/-symmetric, and .R; S/-skew symmetric ma- trices, and block circulants ŒAs˛r r;sD0. Having said this,we now proposethat all the papers in our bibliography– including our own – are based on an unnecessarily restrictive assumption; namely, that the spectra of the matrices R and S that are used to define the symmetries consist of a set (usually the complete set) of k-th roots of unity for some k 2. In this paper we point out that this assumption is irrelevant and present an alternative approach that eliminates this requirement and exposes a wider class of generalized symmetries if k>2. We extend our results in [21] and [23] to this larger class of matrices. 2 Preliminary considerations Throughout the rest of this paper, 1 mm R P diag. 0Im ; 1Im ;:::; k1Im /P C (5) D 0 1 k1 2 and 1 nn S Q diag. 0In ; 1In ;:::; k1In /Q C ; (6) D 0 1 k1 2 where 0, 1,..., k1 are distinct complex numbers, except when there is an explicit statement to the contrary. We define 1 R P diag .0/Im ; .1/Im ;:::; .k1/Im P D 0 1 k1 and 1 S Q diag. .0/In ; .1/In ;:::; .k1/In /Q ; D 0 1 k1 where Zk Zk. We canW partition! P P0 P1 Pk1 ; Q Q0 Q1 Qk1 ; (7) D D P0 Q0 P Q 1 1 1 1 P 2 b: 3 and Q 2 b: 3 ; (8) D : D : 6 b 7 6 b 7 6 7 6 7 6 Pk1; 7 6 Qk1; 7 where 4 5 4 5 b b mmr mr m Pr C ; Pr C ; Pr Ps ırsIm ; 0 r; s k 1; (9) 2 2 D r Ä Ä nnr nr n Qr C ; Qr C ; and Qr Qs ırsIn ; 0 r; s k 1: (10) 2 2b b D r Ä Ä We can now write b b k1 k1 R `P`P`; R .`/P`P`; (11) D D D D X` 0 X` 0 k1 b k1 b S `Q`Q`; and S .`/Q`Q`: (12) D D D D X` 0 X` 0 b b 4 William F. Trench Definition 1 In general, if U Cmm, V Cnn, and A Cmn, we say that A is 2 2 2 mn .U; V /-commutative if UA AV . In particular, we say that A C is .R;S /- D 2 commutative if RA AS . If is the identity (i.e., RA AS), we say that A is .R; S/-commutative.D If A, R Cnn and RA AR, weD say – as usual – that A commutes with R. 2 D 3 Necessary and sufficient conditions for .R;S/-commutativity mn Theorem 1 A C is .R;S /-commutative if and only if 2 k1 1 mr ns A P ŒCrs Q ; where Crs C (13) D r;sD0 2 Á and Crs 0 if r .s/; 0 r; s k 1: (14) D ¤ Ä Ä PROOF. Any A Cm n can be written as in (13) with C P 1AQ partitioned as indicated. If 2 D D diag 0Im ; 1Im ;:::; k1Im D 0 1 k1 and D diag .0/In ; .1/In ;:::; .k1/In ; D 0 1 k1 then 1 1 1 k1 1 RA .PDP /.P CQ / PDCQ P Œ r Crs Q D D D r;sD0 Á and 1 1 1 k1 1 AS .P CQ /.QD Q / P CD Q P Œ .s/Crs Q : D D D r;sD0 Á Therefore RA AS if and only if . r .s//Crs 0, 0 r; s k 1, which is D D Ä Ä equivalent to (14), since 0, 1,..., k1 are distinct. The following theorem is a convenient reformulation of Theorem 1. mn Theorem 2 A C is .R;S /-commutative if and only if 2 k1 m.`/ n` A P.`/F`Q` with F` C ; 0 ` k 1; (15) D 2 Ä Ä D X` 0 b in which case F` P.`/AQ`; 0 ` k 1; (16) D Ä Ä and b k1 RA AS .`/P.`/F`Q` (17) D D D X` 0 for arbitrary 0, 1,..., k1. b .R;S /-commutative matrices 5 PROOF. From (13), an arbitrary A Cm n can be written as 2 k1 k1 A Pr CrsQ`: (18) D sD0 rD0 X X b From Theorem 1, A is .R;S /-commutative if and and only if Crs 0 if r .s/, in m.`/ n` D ¤ which case (18) reduces to (15) with F` C.`/;` C . From (10) and (15), D 2 AQ` P.`/F`, 0 ` k 1, so (9) with r .`/ implies (16). Eqns. (9)–(12) and (15)D imply (17).Ä Ä D Example 1 If is the permutation 012345 .0; 1; 3/.2; 4/.5/; D 134025 D  à then (15) becomes A P1F0Q0 P3F1Q1 P4F2Q2 P0F3Q3 P2F4Q4 P5F5Q5; D C C C C C with b b b b b b m1n0 m3n1 m4n2 F0 C ; F1 C ; F2 C ; 2 2 2 m0n3 m2n4 m5n5 F3 C ; F4 C ; F5 C ; 2 2 2 and RA AS 1P1F0Q0 3P3F1Q1 4P4F2Q2 0P0F3Q3 2P2F4Q4 5P5F5Q5 D D C C C C C for arbitrary 0,..., 5.b b b b b b Example 2 If 012345 D 210120  à (which is not a permutation), then (15) becomes A P2F0Q0 P1F1Q1 P0F2Q2 P1F3Q3 P2F4Q4 P0F5Q5; D C C C C C with b b b b b b m2n0 m1n1 m0n2 F0 C ; F1 C ; F2 C ; 2 2 2 m1n3 m2m4 m0n5 F3 C ; F4 C ; F5 C ; 2 2 2 and RA AS 2P2F0Q0 1P1F1Q1 0P0F2Q2 1P1F3Q3 2P2F4Q4 0P0F5Q5 D D C C C C C for arbitrary 0,..., 5.b b b b b b 6 William F.
Load more

Recommended publications
  • Some Combinatorial Properties of Centrosymmetric Matrices* Allan B
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Some Combinatorial Properties of Centrosymmetric Matrices* Allan B. G-use Mathematics Department University of San Francisco San Francisco, California 94117 Submitted by Richard A. Bruaidi ABSTRACT The elements in the group of centrosymmetric fl X n permutation matrices are the extreme points of a convex subset of n2-dimensional Euclidean space, which we characterize by a very simple set of linear inequalities, thereby providing an interest- ing solvable case of a difficult problem posed by L. Mirsky, as well as a new analogue of the famous theorem on doubly stochastic matrices due to G. Bid&off. Several further theorems of a related nature also are included. INTRODUCTION A square matrix of non-negative real numbers is doubly stochastic if the sum of its entries in any row or column is equal to 1. The set 52, of doubly stochastic matrices of size n X n forms a bounded convex polyhedron in n2-dimensional Euclidean space which, according to a classic result of G. Birkhoff [l], has the n X n permutation matrices as its only vertices. L. Mirsky, in his interesting survey paper on the theory of doubly stochastic matrices [2], suggests the problem of developing refinements of the Birkhoff theorem along the following lines: Given a specified subgroup G of the full symmetric group %,, of the n! permutation matrices, determine the poly- hedron lying within 3, whose vertices are precisely the elements of G. In general, as Mirsky notes, this problem appears to be very difficult.
    [Show full text]
  • On Some Properties of Positive Definite Toeplitz Matrices and Their Possible Applications
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector On Some Properties of Positive Definite Toeplitz Matrices and Their Possible Applications Bishwa Nath Mukhexjee Computer Science Unit lndian Statistical Institute Calcutta 700 035, Zndia and Sadhan Samar Maiti Department of Statistics Kalyani University Kalyani, Nadia, lndia Submitted by Miroslav Fiedler ABSTRACT Various properties of a real symmetric Toeplitz matrix Z,,, with elements u+ = u,~_~,, 1 Q j, k c m, are reviewed here. Matrices of this kind often arise in applica- tions in statistics, econometrics, psychometrics, structural engineering, multichannel filtering, reflection seismology, etc., and it is desirable to have techniques which exploit their special structure. Possible applications of the results related to their inverse, determinant, and eigenvalue problem are suggested. 1. INTRODUCTION Most matrix forms that arise in the analysis of stationary stochastic processes, as in time series data, are of the so-called Toeplitz structure. The Toeplitz matrix, T, is important due to its distinctive property that the entries in the matrix depend only on the differences of the indices, and as a result, the elements on its principal diagonal as well as those lying within each of its subdiagonals are identical. In 1910, 0. Toeplitz studied various forms of the matrices ((t,_,)) in relation to Laurent power series and called them Lforms, without assuming that they are symmetric. A sufficiently well-structured theory on T-matrix LINEAR ALGEBRA AND ITS APPLICATIONS 102:211-240 (1988) 211 0 Elsevier Science Publishing Co., Inc., 1988 52 Vanderbilt Ave., New York, NY 10017 00243795/88/$3.50 212 BISHWA NATH MUKHERJEE AND SADHAN SAMAR MAITI also existed in the memoirs of Frobenius [21, 221.
    [Show full text]
  • New Classes of Matrix Decompositions
    NEW CLASSES OF MATRIX DECOMPOSITIONS KE YE Abstract. The idea of decomposing a matrix into a product of structured matrices such as tri- angular, orthogonal, diagonal matrices is a milestone of numerical computations. In this paper, we describe six new classes of matrix decompositions, extending our work in [8]. We prove that every n × n matrix is a product of finitely many bidiagonal, skew symmetric (when n is even), companion matrices and generalized Vandermonde matrices, respectively. We also prove that a generic n × n centrosymmetric matrix is a product of finitely many symmetric Toeplitz (resp. persymmetric Hankel) matrices. We determine an upper bound of the number of structured matrices needed to decompose a matrix for each case. 1. Introduction Matrix decomposition is an important technique in numerical computations. For example, we have classical matrix decompositions: (1) LU: a generic matrix can be decomposed as a product of an upper triangular matrix and a lower triangular matrix. (2) QR: every matrix can be decomposed as a product of an orthogonal matrix and an upper triangular matrix. (3) SVD: every matrix can be decomposed as a product of two orthogonal matrices and a diagonal matrix. These matrix decompositions play a central role in engineering and scientific problems related to matrix computations [1],[6]. For example, to solve a linear system Ax = b where A is an n×n matrix and b is a column vector of length n. We can first apply LU decomposition to A to obtain LUx = b, where L is a lower triangular matrix and U is an upper triangular matrix.
    [Show full text]
  • Packing of Permutations Into Latin Squares 10
    PACKING OF PERMUTATIONS INTO LATIN SQUARES STEPHAN FOLDES, ANDRAS´ KASZANYITZKY, AND LASZL´ O´ MAJOR Abstract. For every positive integer n greater than 4 there is a set of Latin squares of order n such that every permutation of the numbers 1,...,n appears exactly once as a row, a column, a reverse row or a reverse column of one of the given Latin squares. If n is greater than 4 and not of the form p or 2p for some prime number p congruent to 3 modulo 4, then there always exists a Latin square of order n in which the rows, columns, reverse rows and reverse columns are all distinct permutations of 1,...,n, and which constitute a permutation group of order 4n. If n is prime congruent to 1 modulo 4, then a set of (n − 1)/4 mutually orthogonal Latin squares of order n can also be constructed by a classical method of linear algebra in such a way, that the rows, columns, reverse rows and reverse columns are all distinct and constitute a permutation group of order n(n − 1). 1. Introduction A permutation is a bijective map f from a finite, n-element set (the domain of the permutation) to itself. When the domain is fixed, or it is an arbitrary n-element set, the group of all permutations on that domain is denoted by Sn (full symmetric group). If the elements of the domain are enumerated in some well defined order as z1,...,zn, then the sequence f(z1),...,f(zn) is called the sequence representation of the permutation f.
    [Show full text]
  • On the Normal Centrosymmetric Nonnegative Inverse Eigenvalue Problem3
    ON THE NORMAL CENTROSYMMETRIC NONNEGATIVE INVERSE EIGENVALUE PROBLEM SOMCHAI SOMPHOTPHISUT AND KENG WIBOONTON Abstract. We give sufficient conditions of the nonnegative inverse eigenvalue problem (NIEP) for normal centrosymmetric matrices. These sufficient condi- tions are analogous to the sufficient conditions of the NIEP for normal matrices given by Xu [16] and Julio, Manzaneda and Soto [2]. 1. Introduction The problem of finding necessary and sufficient conditions for n complex numbers to be the spectrum of a nonnegative matrix is known as the nonnegative inverse eigenvalues problem (NIEP). If the complex numbers z1,z2,...,zn are the eigen- values of an n n nonnegative matrix A then we say that the list z1,z2,...,zn is realizable by A×. The NIEP was first studied by Suleimanova in [14] and then several authors had been extensively studied this problem for examples in [1,4–7,9,11–15]. There are related problems to the NIEP when one also want the desired non- negative matrices to be symmetric, persymmetric, bisymmetric or circulant, etc. For example, if one want to find a necessary and sufficient condition for a list of n numbers to be realized by a symmetric nonnegative matrix then this problem is called the NIEP for symmetric matrices or shortly the SNIEP (symmetric nonneg- ative inverse eigenvalue problem). In this paper, we are interested in the NIEP for normal centrosymmetric matrices. There are several results of the NIEP for normal matrices. The following theorem due to Xu [16] gives a sufficient condition of the NIEP for normal matrices. Theorem 1.1. Let λ0 0 λ1 ..
    [Show full text]
  • On the Logarithms of Matrices with Central Symmetry
    J. Numer. Math. 2015; 23 (4):369–377 Chengbo Lu* On the logarithms of matrices with central symmetry Abstract: For computing logarithms of a nonsingular matrix A, one well known algorithm named the inverse scaling and squaring method, was proposed by Kenney and Laub [14] for use with a Schur decomposition. In this paper we further consider (the computation of) the logarithms of matrices with central symmetry. We rst investigate the structure of the logarithms of these matrices and then give the classications of their log- arithms. Then, we develop several algorithms for computing the logarithms. We show that our algorithms are four times cheaper than the standard inverse scaling and squaring method for matrices with central symmetry under certain conditions. Keywords: Logarithm, central symmetry, Schur decomposition, reduced form, primary function. MSC 2010: 65F30, 65F15 DOI: 10.1515/jnma-2015-0025 Received January 28, 2013; accepted May 24, 2014 1 Introduction Matrices with central symmetry arise in the study of certain types of Markov processes because they turn out to be the transition matrices for the processes [6, 20]. Such matrices play an important role in a number of areas such as pattern recognition, antenna theory, mechanical and electrical systems, and quantum physics [8]. These matrices arise also in spectral methods in boundary value problems (BVPs) [5]. For recent years the properties and applications of them have been extensively investigated [5, 6, 8, 13, 18, 20]. Logarithms of matrices arise in various contexts. For example [1, 4, 7, 19], for a physical system governed by a linear dierential equation of the form dy Xy dt where X is an unknown matrix.
    [Show full text]
  • Preconditioners for Symmetrized Toeplitz and Multilevel Toeplitz Matrices∗
    PRECONDITIONERS FOR SYMMETRIZED TOEPLITZ AND MULTILEVEL TOEPLITZ MATRICES∗ J. PESTANAy Abstract. When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The precondi- tioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments. Key words. Toeplitz matrix, multilevel Toeplitz matrix, symmetrization, preconditioning, Krylov subspace method AMS subject classifications. 65F08, 65F10, 15B05, 35R11 1. Introduction. Linear systems (1.1) Anx = b; n×n n where An 2 R is a Toeplitz or multilevel Toeplitz matrix, and b 2 R arise in a range of applications. These include the discretization of partial differential and integral equations, time series analysis, and signal and image processing [7, 27]. Additionally, demand for fast numerical methods for fractional diffusion problems| which have recently received significant attention|has renewed interest in the solution of Toeplitz and Toeplitz-like systems [10, 26, 31, 32, 46]. Preconditioned iterative methods are often used to solve systems of the form (1.1). When An is Hermitian, CG [18] and MINRES [29] can be applied, and their descrip- tive convergence rate bounds guide the construction of effective preconditioners [7, 27]. On the other hand, convergence rates of preconditioned iterative methods for non- symmetric Toeplitz matrices are difficult to describe.
    [Show full text]
  • Stein's Method and the Rank Distribution of Random Matrices Over
    The Annals of Probability 2015, Vol. 43, No. 3, 1274–1314 DOI: 10.1214/13-AOP889 c Institute of Mathematical Statistics, 2015 STEIN’S METHOD AND THE RANK DISTRIBUTION OF RANDOM MATRICES OVER FINITE FIELDS By Jason Fulman1 and Larry Goldstein2 University of Southern California With Qq,n the distribution of n minus the rank of a matrix chosen uniformly from the collection of all n × (n + m) matrices over the finite field Fq of size q ≥ 2, and Qq the distributional limit of Qq,n as n →∞, we apply Stein’s method to prove the total variation bound 1 3 ≤ kQq,n −Qqk ≤ . 8qn+m+1 TV qn+m+1 In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices. 1. Introduction. We study the distribution of the rank for various en- sembles of random matrices over finite fields. To give a flavor of our results, let Mn be chosen uniformly from all n × (n + m) matrices over the finite field Fq of size q ≥ 2. Letting Qq,n = n − rank(Mn), it is known (page 38 of [3]) that for all k in Un = {0,...,n}, P (Qq,n = k)= pk,n (1) n+m i n i 1 i=1 (1 − 1/q ) i=k+1(1 − 1/q ) where pk,n = . qk(m+k) n−k i m+k i Q i=1 (1 − 1/q )Q i=1 (1 − 1/q ) N Clearly, for any fixed k ∈ 0, the collectionQ of nonnegativeQ integers, ∞ i 1 i=k+1(1 − 1/q ) (2) lim pk,n = pk where pk = .
    [Show full text]
  • Pseudo-Centrosymmetric Matrices, with Applications to Counting Perfect
    Pseudo-centrosymmetric matrices, with applications to counting perfect matchings Christopher R. H. Hanusa a aDepartment of Mathematical Sciences, Binghamton University, Binghamton, New York, 13902-6000 [email protected] Abstract We consider square matrices A that commute with a fixed square matrix K, both with entries in a field F not of characteristic 2. When K2 = I, Tao and Yasuda defined A to be generalized centrosymmetric with respect to K. When K2 = I, − we define A to be pseudo-centrosymmetric with respect to K; we show that the determinant of every even-order pseudo-centrosymmetric matrix is the sum of two squares over F , as long as 1 is not a square in F . When a pseudo-centrosymmetric − matrix A contains only integral entries and is pseudo-centrosymmetric with respect to a matrix with rational entries, the determinant of A is the sum of two inte- gral squares. This result, when specialized to when K is the even-order alternating exchange matrix, applies to enumerative combinatorics. Using solely matrix-based methods, we reprove a weak form of Jockusch’s theorem for enumerating perfect matchings of 2-even symmetric graphs. As a corollary, we reprove that the number of domino tilings of regions known as Aztec diamonds and Aztec pillows is a sum of two integral squares. Key words: centrosymmetric, anti-involutory, pseudo-centrosymmetric, determinant, alternating centrosymmetric matrix, Jockusch, 2-even symmetric graph, Kasteleyn-Percus, domino tiling, Aztec diamond, Aztec pillow arXiv:math/0609622v2 [math.CO] 9 Jul 2007 1 Introduction This article is divided into two halves; results concerning certain types of matrices from the first half apply to enumerative combinatorics in the second half.
    [Show full text]
  • Inverses of 2 × 2 Block Matrices
    An Intematiomd Journal computers & mathematics with applications PERGAMON Computers and Mathematics with Applications 43 (2002) 119-129 www.eisevier.com/locate/camwa Inverses of 2 × 2 Block Matrices TZON-TZER LU AND SHENG-HUA SHIOU Department of Applied Mathematics National Sun Yat-sen University Kaohsiung, Taiwan 80424, R.O.C. ttluCmath, nsysu, edu. tv (Received March 2000; revised and accepted October P000) Abstract--ln this paper, the authors give explicit inverse formulae for 2 x 2 block matrices with three differentpartitions. Then these resultsare applied to obtain inversesof block triangular matrices and various structured matrices such as Hamiltonian, per-Hermitian, and centro-Hermitian matrices. (~) 2001 Elsevier Science Ltd. All rights reserved. Keywords--2 x 2 block matrix, Inverse matrix, Structured matrix. 1. INTRODUCTION This paper is devoted to the inverses of 2 x 2 block matrices. First, we give explicit inverse formulae for a 2 x 2 block matrix D ' (1.1) with three different partitions. Then these results are applied to obtain inverses of block triangular matrices and various structured matrices such as bisymmetric, Hamiltonian, per-Hermitian, and centro-Hermitian matrices. In the end, we briefly discuss the completion problems of a 2 x 2 block matrix and its inverse, which generalizes this problem. The inverse formula (1.1) of a 2 x 2 block matrix appears frequently in many subjects and has long been studied. Its inverse in terms of A -1 or D -1 can be found in standard textbooks on linear algebra, e.g., [1-3]. However, we give a complete treatment here. Some papers, e.g., [4,5], deal with its inverse in terms of the generalized inverse of A.
    [Show full text]
  • Document Downloaded From: This Paper Must Be Cited As: the Final
    Document downloaded from: http://hdl.handle.net/10251/119992 This paper must be cited as: Lebtahi Ep-Kadi-Hahifi, L.; Romero Martínez, JO.; Thome, N. (2018). Further results on generalized centro-invertible matrices. Numerical Algorithms. 80(4):1309-1328. https://doi.org/10.1007/s11075-018-0528-9 The final publication is available at https://doi.org/10.1007/s11075-018-0528-9 Copyright Springer-Verlag Additional Information Further results on generalized centro-invertible matrices Leila Lebtahi∗ Oscar´ Romero† N´estor Thome‡ Abstract This paper deals with generalized centro-invertible matrices intro- duced by the authors in [Generalized centro-invertible matrices with ap- plications, Applied Mathematics Letters, 38, 106–109, 2014]. As a first result, we state the coordinability between the classes of involutory ma- trices, generalized centro-invertible matrices, and {K}-centrosymmetric matrices. Then, some characterizations of generalized centro-invertible matrices are obtained. A spectral study of generalized centro-invertible matrices is given. In addition, we prove that the sign of a generalized centro-invertible matrix is {K}-centrosymmetric and that the class of gen- eralized centro-invertible matrices is closed under the matrix sign func- tion. Finally, some algorithms have been developed for the construction of generalized centro-invertible matrices. AMS Classification: 15A09, 15A18 Keywords: Centro-symmetric matrices; centro-invertible matrices; spectral analysis; inverse problem; matrix sign function. 1 Introduction × Centrosymmetric matrices are those complex matrices A ∈ Cn n such that n×n AJn = JnA where Jn ∈ C is the matrix with 1’s on the secondary diagonal and 0’s otherwise. An important property of centrosymmetric matrices is that if A is a centrosymmetric matrix with γ linearly independent eigenvectors, then γ linearly independent eigenvectors of A can be chosen to be symmetric or skew- centrosymmetric.
    [Show full text]
  • Involutions and Generalized Centrosymmetric and Skew-Centrosymmetric Matrices
    CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 15, Number 1, Spring 2007 INVOLUTIONS AND GENERALIZED CENTROSYMMETRIC AND SKEW-CENTROSYMMETRIC MATRICES IYAD T. ABU-JEIB ABSTRACT. We present new results and simple short proofs for known theorems about generalized centrosymmetric matri- ces and generalized skew-centrosymmetric (centroskew) matri- ces. We also make a comparison between the identity matrix and a nontrivial involutory matrix K by studying the eigensys- tem of K + M, where M is a structured matrix, with the goal of expressing this system entirely in terms of the corresponding eigensystem of M. Also, properties of some parametric families of the form K + ρM are studied. 1 Introduction Let K be an n n nontrivial (i.e., K = I) involutory matrix, x an n 1 vector, ×and A an n n matrix.6 A is called K-centrosymmetric (or× K-symmetric) if AK =×KA and K-skew- centrosymmetric (or K-skew-symmetric) if AK = KA; x is called K-symmetric if Kx = x and K-skew-symmetric if−Kx = x. K- centrosymmetric and K-skew-centrosymmetric matrices were studied− in [5, 6, 7]. Properties of the sum of two matrices have been studied by many researchers in many contexts with the objective of finding connections between the eigensystem of the summands and the eigensystem of the sum. In particular, the eigenvalues of the sum of the identity matrix I and another matrix is one of the first sums that one encounters in elementary linear algebra. A very useful relative of the identity is the counteridentity J, which is obtained from the identity by reversing the order of its columns.
    [Show full text]