Iterative Solution to a System of Matrix Equations

Abstract EN

An efficient iterative algorithm is presented to solve a system of linear matrix equations A1X1B1+A2X2B2=E, C1X1D1+C2X2D2=F with real matrices X1 and X2.

By this iterative algorithm, the solvability of the system can be determined automatically.

When the system is consistent, for any initial matrices X10 and X20, a solution can be obtained in the absence of roundoff errors, and the least norm solution can be obtained by choosing a special kind of initial matrix.

In addition, the unique optimal approximation solutions X^1 and X^2 to the given matrices X~1 and X~2 in Frobenius norm can be obtained by finding the least norm solution of a new pair of matrix equations A1X¯1B1+A2X¯2B2=E¯, C1X¯1D1+C2X¯2D2=F¯, where E¯=E-A1X~1B1-A2X~2B2, F¯=F-C1X~1D1-C2X~2D2.

The given numerical example demonstrates that the iterative algorithm is efficient.

Especially, when the numbers of the parameter matrices A1,A2,B1,B2,C1,C2,D1,D2 are large, our algorithm is efficient as well.