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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. From observations of the state vector y t . If y 0 y then y t eXt y , = 0 0 where the exponential of a matrix is dened by ( ) ( ) = ( ) = ∞ Xk eX . (1.1) k! k=0 By observing y at t 1 for initial states consisting of= theH columns of the identity matrix, we obtain the matrix A eX. Under certain conditions on A and X, we can then solve for X as X log A . This raises the question of how to compute= a logarithm of a matrix. = The standard inverse scaling and squaring method was proposed by Kenney= ( and) Laub [14] for use with a Schur decomposition. It was further developed by Cheng, Higham, Kenney, and Laub [4] in a transformation- free form that takes as a parameter the desired accuracy. The rich variety of methods for computing the loga- rithms of matrices, with their widely diering numerical stability properties, is an interesting subject of study. For these reasons, many authors became interested in the matrix logarithms [2, 9, 11, 15]. Although the theory of matrix logarithms is rather complicated, simplications occur for certain classes of matrices. Consider, for example, symmetric positive denite matrices [16], symplectic matrices and orthogonal matrices [3, 10], etc. Here, our interest is focused on the logarithms of the matrices with a central symmetric structure. This paper is organized as follows. In next section, we review some basic properties of centrosymmetric (centro- *Corresponding Author: Chengbo Lu: Department of Mathematics, Lishui University, Lishui 323000, China. E-mail: [email protected] Authenticated | [email protected] author's copy Download Date | 12/11/15 9:30 AM 370 Ë C. B. Lu, On the logarithms of matrices hermitian) matrices. In Section 3 we investigate the structures of the matrix logarithms, then give the classi- cations of them and develop some algorithms for computing the logarithms in Section 4. We end up with some conclusions and future work. 2 Preliminaries Throughout this paper we denote the set of all n n complex matrices by n×n and the set of all real matrices by n×n. Recall that a matrix A is said to be centrosymmetric× if A JAJ; skewℂ centrosymmetric if A JAJ; centrohermitianℝ if A JAJ, where A denotes the elementwise conjugate of the matrix A and J is the exchange matrix with ones on the cross diagonal (bottom left to top right)= and zeros elsewhere. = − We rst review the= structurē and̄ reducibility of centrosymmetric (centrohermitian) matrices. All the for- mulas become slightly more complicated when n is odd; for simplicity, we restrict our attention to the case of even n 2m. Using an appropriate partition, the central symmetric characters of all n n centrosymmetric matrices and skew= centrosymmetric matrices can be always, respectively, expressed as follows: × BJ CJ A m m if A is centrosymmetric (2.1) CJm BJm and = ¡ B J CJ A m m if A is skew centrosymmetric (2.2) C Jm BJm − where B and C are square matrices= of order m ¡n 2. − Denote by = 2/ I I P m m (2.3) 2 Jm Jm $ where Im is a mth unit matrix. Then we have the= following well-known¡ results. − Lemma 2.1. Let P be dened as in (2.3). Then, (i) A is an n n centrosymmetric matrix if and only if T × P AP diag M, N (2.4) holds, where M B Jm C and N B Jm C. = ( ) (ii) A is an n n skew centrosymmetric matrix if and only if = + = − E × PT AP (2.5) F = ¡ holds, where E B Jm C and F B Jm C. Similarly, any n n centrohermitian matrix always takes the following form: = + = − BJ CJ × A m m . (2.6) CJm BJm ̄ Denote by = ̄ ¡ 2 I iI Q m m (2.7) 2 Jm iJm $ where i is the imaginary unit. Then we have the= following result¡ (see, e.g., [13]). − Authenticated | [email protected] author's copy Download Date | 12/11/15 9:30 AM C. B. Lu, On the logarithms of matrices Ë 371 Lemma 2.2. Let Q be dened as in (2.7). Then, A is an n n centrohermitian matrix if and only if H Q AQ ×RA (2.8) is an n n real matrix, where = Re B Jm C Im B Jm C RA × Im B Jm C Re B Jm C ( + ) − ( + ) with Re T and Im T denoting the real= and imaginary parts of the matrix ¡T, respectively. ( − ) ( − ) We will say that the matrix on the right side of (2.4), (2.5), (2.8) is the reduced form of the matrix A in (2.1), (2.2), ( ) ( ) (2.6), respectively. We end up this section with a well known result regarding the logarithms of a nonsingular matrix and two denitions which will be used in the following sections. Lemma 2.3 (see [12]). Let the nonsingular matrix A n×n have the Jordan canonical form Z−1AZ J diag J1, J2,..., Jp , and let s p be the number of distinct eigenvalues of A. Let ∈ ℂ m 1 = = f λ f λ f k (λk) ( ) ⩽ k k mk 1 ! ὔ (( −−) ) .. (j) (j) f λk . Lk Lk λk ( ) ( ) ⋅ ⋅ ⋅ .. f λk , ( ) ὔ - = ( ) = f λk ( ) where f x log x . Then eX A has a countable innity of solutions that are primary functions of A, given by < ( ) = j j j ( 1) ( 2) ( p) −1 ( ) = ( ) = Xj Z diag L1 , L2 ,..., Lp Z corresponding to all possible choices of integers= j1(,..., jp, subject to the) constraint that ji jk whenever λi λk. If s p, A has nonprimary logarithms. They form parametrized families j j j = = ( 1) ( 2) ( p) −1 −1 < Xj U ZUdiag L1 , L2 ,..., Lp U Z where jk is an arbitrary integer, U is( an) = arbitrary nonsingular( matrix that) commutes with J, and for each j there exist i and k, depending on j, such that λi λk while ji jk. Among the logarithms of a matrix, the principal logarithm is distinguished by its usefulness in theory and = ̸= application. Denote by −1 the closed negative axis. Denition 2.1 (see [11], principal logarithm). Let A n×n have no eigenvalues on −1. There is a unique ℝ logarithm X of A all of whose eigenvalues lie in the strip z : π Im z π . We refer to X as the principal logarithm of A and write X log A . ∈ ℂ ℝ { − < ( ) < } Denition 2.2 (see [12]). Let f be dened on the spectrum of A n×n and let ψ be the minimal polyno- = ( ) s mial of A. Then f A : p A , where P is a polynomial of degree less than ni deg ψ that satises the i=1 interpolation conditions ∈ ℂ ( ) = ( ) ∑ = j j p( ) λi f ( ) λi , j 0, 1,..., ni 1, i 1, 2,..., s. There is a unique such p and( it is) = known( ) as the= Hermite interpolating− = polynomial. This paper will deal with the function f λ log λ , which is clearly dened on the spectrum of a nonsingular matrix A. ( ) = ( ) 3 Matrix logarithms In this section we present some new results which characterize the logarithms of nonsingular matrices with central symmetry. Authenticated | [email protected] author's copy Download Date | 12/11/15 9:30 AM 372 Ë C. B. Lu, On the logarithms of matrices 3.1 Logarithms of centrosymmetric matrices For the special structure of the centrosymmetric matrices, one may ask whether they have logarithms which are also centrosymmetric or skew centrosymmetric. We have some answers to this question. Theorem 3.1. Let A n×n be a centrosymmetric matrix. Then, A has a centrosymmetric logarithm if and only if each of M and N in (2.4) admits a logarithm. ∈ ℂ Proof. By assumption, we have that A is centrosymmetric. Then, by Lemma 2.1, A has the reduced form (2.4). Assume that A has a centrosymmetric logarithm, denoted by X. By Lemma 2.1 again, we have PT XP diag X , X (3.1) â⇒ 1 2 where P is dened by the relation (2.3).
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