Noin~. ~ Matrix Decompositions Using Displacement Rank and Classes of Commutative Matrix Algebras

Using the notion of displacement rank, we look for a unifying approach to representations of a matrix A as sums of products of matrices belonging to commutative matrix algebras. These representations are then considered in case A is the inverse of a Toeplitz or a Toeplitz plus Hankel matrix. Some well-known decomposition formulas for A (Gohberg-Semencul or Kailath et al., Gader, Bini-Pan, and Gohberg-Olshevsky) turn out to be special cases of the above representations. New formulas for A in terms of algebras of symmetric matrices are studied, and their computational aspects are discussed. 1. I N T R O D U C T I O N It is well known that the notion of displacement rank underlies many algorithms for solving Toeplitz systems of equations and that the same notion can be used to extend algorithms for Toeplitz matrices to other classes of matrices A [4, 5, 7, 10, 13-22, 24]. The main idea consists in looking for an LINEAR ALGEBRA AND ITS APPLICATIONS 229:49-99 (1995) © Elsevier Scienc~e Inc., 1995 0024-3795/95/$9.50 655 Avenue of the Americas, New York, NY 10010 SSDI 0024-3795(93)00347-3 50 CARMINE DI FIORE AND PAOLO ZELLINI operator ~ which transforms A into a low rank matrix ~ (A) such that one could easily recover A from its image ~(A). A consequent expression for A is then obtained, which depends on the rank of ~ (A) and is formulated in terms of (possibly) a few simple structured matrices. The classical GohbergSemencul [17] or Kailath et al. [22] formulas, the circulant type formulas of Gader [14] (see also Bini [7]), the e-circulant type formulas of Gohberg and Olshevsky [16], and other known formulas involving a special algebra ~" of matrices [7, 10] are all examples of the above technique. These formulas for A are then useful for solving computational problems--for example, a linear system--by means of any of a number of fast transforms (typically the FFT). In the present paper we look for a unifying approach by exploiting a class of commutative matrix algebras (Section 2) containing, as particular instances, all algebras considered in the literature (7, circulant, e-circulant, Toeplitz triangular), with the sole exception of the group algebras different from circulant matrices used in [14]. This class of algebras is constructed with a technique which is similar, in spirit, to that used by Bapat and Sunder in their paper on hypergroups of matrices [6]. By this general approach we are able to formulate a decomposition theorem (Theorem 3.1) whose corollaries give the well-known splits for A based on the previously mentioned algebras. New decomposition formulas for A are then obtained involving whole classes of algebras instead of singular algebras of matrices (Section 3: in particular Theorems 3.2 and 3.3). In Section 4 are listed, as particular instances, some interesting formulas for T -1 and (T + H ) -1 where T is a Toeplitz and T + H is a Toeplitz plus Hankel matrix [18-20]. Especially in the case of (T + H ) -1, some of these formulas appear to be particularly simple and effective, as they involve only a few products of elements of the same algebra ~'. Some computational aspects of these formulas are then investigated in Section 5. All previous results are obtained using, as ~(A) , the commutator ~ x(A) = A X XA for different choices of X, depending on the matrix algebra involved. In fact, ~x(A) turns out to be the most natural operator, as the matrix algebras considered throughout the paper are commutative. For the sake of completeness we state Propositions 4.3 and 4.4, in Section 4, which show that for some convenient choices of X the images ~x(T -1) or ~x((T + H)-I) can always be expressed in terms of a number of columns or rows and columns of, respectively, T -1 and (T + H ) -1. 2. A CLASS OF ALGEBRAS OF MATRICES In this section we shall introduce a class of algebras of n x n matrices over C, using a constructive criterion similar, in some ways, to that proposed by Bapat and Sunder in their paper on hypergroups of matrices [6]. This class MATRIX DECOMPOSITIONS USING DISPLACEMENT RANK 51 of algebras will be exploited to write an arbitrary square matrix over a ring as sums of matrix products, in the spirit of the literature on rank displacement operators [4, 5, 7, 10, 13-22, 24]. A general approach will follow from which one is able to regain, as special cases, the classical Gohberg-Semencul formulas [17] (or Kailath et al. [22]), the variants proposed by Gader in [14] and by Bini and Pan in [10] (see also Bini [7]), and the Gohberg-Olshevsky formulas [16], Consider the lower Hessenberg matrix X = rll b 1 0 r2i rz2 bz