Polynomial Solutions to the Matrix Equation X−AXTB=C

Abstract

Solutions are constructed for the Kalman-Yakubovich-transpose equation $X-A{X}^{T}B=C$. The solutions are stated as a polynomial of parameters of the matrix equation. One of the polynomial solutions is expressed by the symmetric operator matrix, controllability matrix, and observability matrix. Moreover, the explicit solution is proposed when the Kalman-Yakubovich-transpose matrix equation has a unique solution. The provided approach does not require the coefficient matrices to be in canonical form. In addition, the numerical example is given to illustrate the effectiveness of the derived method. Some applications in control theory are discussed at the end of this paper.