Generalized conjugate direction algorithm for solving the general coupled matrix equations over symmetric matrices

AbstractSymmetric solutions of the linear matrix equations have wide applications in both mechanical and electrical engineering. In this work, an analytic study of the generalized conjugate direction (CD) algorithm for finding the symmetric solution group (X1,X2,...,Xm) of the general coupled matrix equations ∑j=1mAijXjBij=Ci,i=1,2,...,n,$$\sum\limits_{j=1}^{m}A_{ij}X_{j}B_{ij}=C_{i},\qquad i=1,2,...,n, $$ is performed. We show that the generalized CD algorithm can compute the (least Frobenius norm) symmetric solution group of the general coupled matrix equations for any (special) initial symmetric matrix group within a finite number of iterations in the absence of round-off errors. In order to illustrate the effectiveness of the generalized CD algorithm, two numerical examples are finally given.

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