An iterative algorithm for the generalized reflexive solutions of the general coupled matrix equations

作者：Yin, Feng^*; Guo, Ke; Huang, Guang-Xin

来源：JOURNAL OF INEQUALITIES AND APPLICATIONS, 2013, 2013(1): 280.

DOI：10.1186/1029-242X-2013-280

• iterative method
• generalized reflexive solution
• general coupled matrix equations
• optimal approximate solution

The general coupled matrix equations
Sigma(q)(j=1) A(ij)B(ij) = M-ji i = 1,2,..., p
(including the generalized coupled Sylvester matrix equations as special cases) have numerous applications in control and system theory. In this paper, an iterative algorithm is constructed to solve the general coupled matrix equations and their optimal approximation problem over generalized reflexive matrix solution (X-1, X-2,..., X-q). When the general coupled matrix equations are consistent over generalized reflexive matrices, the generalized reflexive solution can be determined automatically by the iterative algorithm within finite iterative steps in the absence of round-off errors. The least Frobenius norm generalized reflexive solution of the general coupled matrix equations can be derived when an appropriate initial matrix group is chosen. Furthermore, the unique optimal approximation generalized reflexive solution ((X) over tilde (1),(X) over tilde (2),...,(X) over tilde (q)) to a givenmatrix group (X-1(0), X-2(0),..., X-q(0)) in Frobenius norm can be derived by finding the least-norm generalized reflexive solution ((X) over tilde (*)(1),(X) over tilde (*)(2),..., (X) over tilde (*)(q)) of the corresponding general coupled matrix equations Sigma(q)(j= 1) Aij (X) over tilde B-j(ij) = (M) over tilde (i), i = 1,2,..., p, where (x) over tilde (j) = X-j -X-j(0), (M) over tilde (i) = M-i -Sigma(q)(j=1) A(ij)X(j)(0) B-ij. A numerical example is given to illustrate the effectiveness of the proposed iterative algorithm.

• 出版日期2013-6-4
• 单位四川轻化工大学; 成都理工大学