A finite iterative method for solving a pair of linear matrix equations ☆
Introduction
Denoted by be the set of real matrices, AT and A† be the transpose and Moore–Penrose generalized inverse of matrix A, respectively. In the space Rm × n, we define an inner product as for all . Then the norm of a matrix A generated by this inner product is, obviously, Frobenius norm and denoted by ∥A∥.
We consider the solution of the pair of linear matrix equations:where and .
We also consider the solution of the matrix optimal approximation problem.where is the given matrix and SE is the solution set of the pair of matrix equation (1).
Since Mitra first gave necessary and sufficient conditions for a pair of individually consistent linear matrix equation (1) to have a common solution and presented the expression for a general common solution when the stated condition holds in Ref. [1], there have been many papers to discuss the matrix equation pair (e.g. [1], [2], [3], [4], [5], [6], [7], [8]). For instance, Van der Woude [3], [4] investigated it over a field in 1987. Özgüler and Akar [5] gave a condition for the solvability of the system over a principle domain in 1991. Wang [7], [8] studied the system over an arbitrary division ring in 1996 and regular ring in 2004, respectively.
The matrix nearness problem (2) occurs frequently in experimental design, see for instance [9]. In recent years, the matrix nearness problems have been studied extensively in many papers. Here may be obtained from experiments, but not satisfy the pair of the matrix equation (1). The nearness problem satisfies the matrix equations pair (1) and is closed to the given matrix in Frobenius norm (may be spectral norm or others).
Using the iterative method to solve the matrix equation can be found in paper [10], [11]. But using this method in a pair of matrix equations cannot be viewed in any paper. In this paper, an efficient iterative method is presented to solve the pair of linear matrix equation (1) for any real matrix X. The suggested iterative method, automatically determines the solvability of equations pair (1). When the pair of equations is consistent, then, for any initial matrix X0, a solution can be obtained within finite iteration steps in the absence of round errors, and the least norm solution can be obtained by choosing a special kind of initial matrix. In addition, using this iterative method, the solution of minimization problem (2) can be obtained by first finding the least norm solution of the new pair of linear matrix equations over unknown real matrix , where . The given numerical examples demonstrate that the iterative method is quite efficient.
Section snippets
Iterative method for solving (1) and (2)
In this section, we first introduce to obtain a solution of the pair of linear matrix equation (1). We then show that if the equations pair (1) are consistent, then, for any initial matrix X0, the sequence of matrix {Xk}, generated by the iterative method, converges to their solution with at most pq iteration steps in the absence of roundoff errors and also show that if we let the above initial matrix be chosen as , where H and are all arbitrary, then the solution X∗ obtained
Numerical examples
In this section, we will give some numerical examples to illustrate our results. All the tests are performed by MATLAB 6.5 and the initial iterative matrix is chosen as . Because of the influence of the error of roundoff, we regard matrix A as zero matrix if . Example 4.1 Given matrices and F as follows:and
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The research of this paper was supported by the National Natural Science Foundation (No. 10371044), PR China, The Science and Technology Commission of Shanghai Municipality through Grant (No. 04JC14031) and the University Young Teacher Sciences Foundation of Anhui Province (No. 2006jql220zd).