Least-squares partially bisymmetric solutions of coupled Sylvester matrix equations accompanied by a prescribed submatrix constraint

The coupled Sylvester matrix equations (CSMEs) {A(1)XB(1)+C1YD1=E-1, A(2)XB(2)+C2YD2=E-2, appear frequently in various fields of mathematics and engineering such as in control systems and signal processing. In this investigation, we establish and analyze the generalized conjugate directions (GCDs) method for solving the CSMEs over partially bisymmetric matrices X and Y with a prescribed submatrix constraint. We show that the GCD method with the arbitrary initial bisymmetric matrices can compute the least-squares partially bisymmetric solutions with a prescribed submatrix constraint within a finite number of iterations in the absence of round-off errors. Numerical examples illustrate the efficiency and simplicity of the GCD method and confirm the theoretical results.

Automated summary

The study focuses on the generalized conjugate directions method for solving coupled Sylvester matrix equations, showing that the method can compute least-squares partially bisymmetric solutions with a prescribed submatrix constraint and converge within a finite number of iterations.