On the solutions of the equation AXB = C under Toeplitz‐like and Hankel matrices constraint

Abstract

In this paper, we are mainly concerned with 2 types of constrained matrix equation problems of the form AXB=C, the least squares problem and the optimal approximation problem, and we consider several constraint matrices, such as general Toeplitz matrices, upper triangular Toeplitz matrices, lower triangular Toeplitz matrices, symmetric Toeplitz matrices, and Hankel matrices. In the first problem, owing to the special structure of the constraint matrix scriptL, we construct special algorithms; necessary and sufficient conditions are obtained about the existence and uniqueness for the solutions. In the second problem, we use von Neumann alternating projection algorithm to obtain the solutions of problem. Then we give 2 numerical examples to demonstrate the effectiveness of the algorithms.

A New Generalized Θ-Inverse vs. Moore-Penrose Structure: A Comparative Control-Oriented Practical Investigation

“…The well-known Moore-Penrose inverse has often been applied in a variety of engineering tasks, including the branches of the control theory, signal processing, electrical networks, pattern recognition, etc. [2], [7]- [14]. Moreover, the original accepted algorithms in the Maple environment providing the Moore-Penrose inverse for any defined nonsquare matrices, have resulted in the interesting starting points for the world multivariable system analysis [15]- [17].…”

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

“…For example, the coupled Sylvester matrix equations often arise in computing an additive decomposition of a generalized transform matrix equations, in computing deflated subspaces of descriptor linear systems, and in problems of reordering eigenvalues of regular matrix pairs. The literature on the approximation of solutions of linear matrix equations is large and is still growing rapidly 13‐22 . By applying the Moore–Penrose generalized inverse and the Kronecker product of matrices, Yuan and Liao proposed the expressions of various least squares solutions with the least norm of the quaternion matrix equation 23 .…”

“…The literature on the approximation of solutions of linear matrix equations is large and is still growing rapidly. [13][14][15][16][17][18][19][20][21][22] By applying the Moore-Penrose generalized inverse and the Kronecker product of matrices, Yuan and Liao proposed the expressions of various least squares solutions with the least norm of the quaternion matrix equation. 23 Chiang presented some useful sufficient conditions for the solvability of the linear matrix equation AX + (X)B = C and derived the expressions of the explicit solutions of the matrix equation.…”