The Chebyshev solution of the linear matrix equationAX+YB=C

Abstract

Summary. In this paper we investigate the properties of the Chebyshev solutions of the linear matrix equation AX + YB = C, where A, B and C are given matrices of dimensions m x r, s • n and m • n, respectively, where r < m and s <n. We separately consider two particular cases. In the first case we assume m=r+l and n=s+l, in the second case r=s=l and m, n are arbitrary. For these two cases, under the assumption that the matrices A and B are full rank, we formulate necessary and sufficient conditions characterizing the Chebyshev solution of AX+ YB= C and we give the formulas for the Chebyshev error. Then, we propose an algorithm which may be applied to compute the Chebyshev solution of AX+YB=C for some particular cases. Some numerical examples are also given.

The Chebyshev solution of certain matrix equations

“…This is not surprising, since such questions are invariable hard, see [9,11,12,1,3]. An important sub-case of the above occurs when r = s = 1, and this is treated at some length in [14]. We propose to investigate this subcase in some detail.…”

“…for all matrices X of size m x 1 and Y of size 1 x n. Thus, as claimed, our treatment encompasses that of [14], in the case r=s= 1 …”

“…In the particular case of the Chebyshev norm, it is also known as the DilibertoStraus algorithm [2]. The methods of [14] are matrix in character, and are insufficient to show that the algorithm converges in general. This is not surprising, since such questions are invariable hard, see [9,11,12,1,3].…”

“…Here, if Z=(zi~ ) is an m x n matrix, then 11 Z I[ ~ = max Izij I. lj The paper [14] deals with exactly this situation, and an algorithm is presented which under strong hypotheses generates a suitable pair Xo0 and Yoo. The algorithm proposed is a version of the alternating algorithm of yon Neumann [13].…”

“…In a series of papers, Zietak [14] has investigated the problem of approximating a matrix C by matrices of the form XA+BY. Here C is a given m x n matrix, A is given r x n matrix, B is a given m x s matrix and we seek matrices X, of size m x r, and Y, of size s x n, which minimise C-XA-BY in some sense.…”