The Inverses of Block Toeplitz Matrices

Journal of Mathematics, Apr 2013

We study the inverses of block Toeplitz matrices based on the analysis of the block cyclic displacement. New formulas for the inverses of block Toeplitz matrices are proposed. We show that the inverses of block Toeplitz matrices can be decomposed as a sum of products of block circulant matrices. In the scalar case, the inverse formulas are proved to be numerically forward stable, if the Toeplitz matrix is nonsingular and well conditioned.

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The Inverses of Block Toeplitz Matrices

The Inverses of Block Toeplitz Matrices Xiao-Guang Lv and Ting-Zhu Huang School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China Received 30 December 2012; Accepted 26 March 2013 Academic Editor: Peter Grabner Copyright © 2013 Xiao-Guang Lv and Ting-Zhu Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We study the inverses of block Toeplitz matrices based on the analysis of the block cyclic displacement. New formulas for the inverses of block Toeplitz matrices are proposed. We show that the inverses of block Toeplitz matrices can be decomposed as a sum of products of block circulant matrices. In the scalar case, the inverse formulas are proved to be numerically forward stable, if the Toeplitz matrix is nonsingular and well conditioned. 1. Introduction Let be an block Toeplitz matrix with blocks of size . We use the shorthand for a block Toeplitz matrix. The block Toeplitz systems arise in a variety of applications in mathematics, scientific computing, and engineering, for instance, image restoration problems in image processing, numerical differential equations and integral equations, time series analysis, and control theory [1–3]. If we want to solve more than one block Toeplitz linear system with the same coefficient matrix, then we usually solve four or so special block linear systems in order to determine the block Toeplitz inverse formula that expresses as the sum of products of block upper and block lower Toeplitz matrices. For example, Van Barel and Bultheel [4] gave an inverse formula for a block Toeplitz matrix and then derived a weakly stable algorithm to solve a block Toeplitz system of linear equations. The special structure of block Toeplitz matrices has resulted in some closed formulas for their inverses. In the scalar case, Gohberg and Semencul [5] have shown that if the st entry of the inverse of a Toeplitz matrix is nonzero, then the first and the last columns of the inverse of the Toeplitz matrix are sufficient to reconstruct . In [6], an inverse formula can be obtained by the solutions of two equations (the so-called fundamental equations), where each right-hand side of them is a shifted column of the Toeplitz matrix. Later, Ben-Artzi and Shalom [7], Labahn and Shalom [8], Huckle [9], Ng et al. [10], and Heinig [11] have studied the Toeplitz matrix inverse formulas when the st entry of the inverse of a Toeplitz matrix is zero. In [12], Cabay and Meleshko presented an efficient algorithm (NPADE) for numerically computing Padé approximants in a weakly stable fashion. As an application of NPADE, it has been shown that it can be used to compute stably, in a weak sense, the inverse of a Hankel or Toeplitz matrix. When , additional problems are encountered in obtaining the inverse formula of a block Toeplitz matrix. A well-known formula of Gohberg and Heinig can construct , provided that the first and last columns together with the first and last rows of the inverse are known [13]. In [14], a set of new formulas for the inverse of a block Hankel or block Toeplitz matrix is given by Labahn et al. The formulas are expressed in terms of certain matrix Padé forms, which approximate a matrix power series associated with the block Hankel matrix. We refer the reader to [15, 16] for the computation of Padé-Hermite and simultaneous Padé systems in detail. In [7], Ben-Artzi and Shalom have proved that each inverse of a Toeplitz matrix can be constructed via three of its columns, and thus, a parametrization of the set of inverses of Toeplitz matrices is obtained. Then they generalized these results to block Toeplitz matrices; see [17]. In [18], Gemignani has shown that the representation of relies upon a strong structure-preserving property of the Schur complements of the nonsingular leading principal submatrices of a certain generalized Bezoutian of matrix polynomials. In this paper, we focus our attention to the inverses of block Toeplitz matrices with the help of the block cyclic displacement. In [19], Ammar and Gader have shown that the inverse of a Toeplitz matrix can be represented as sums of products of lower triangular Toeplitz matrices and circulant matrices. The derivation of their results is based on the idea of cyclic displacement structure. In [20], Gohberg and Olshevsky also obtained new formulas for representation of matrices and their inverses in the form of sums of products of factor circulant, which are based on the analysis of the factor cyclic displacement of matrices. The results in applications to Toeplitz matrices generalized the Gohberg-Semencul, Ben-Artzi-Shalom, and Heinig-Rost formulas. Motivated by a number of related results on Toepltiz inverse formulas, we study the representation of the inverses of block Toeplitz (...truncated)


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Xiao-Guang Lv, Ting-Zhu Huang. The Inverses of Block Toeplitz Matrices, Journal of Mathematics, 2013, 2013, DOI: 10.1155/2013/207176