ON THE REFLEXIVE SOLUTIONS OF THE MATRIX EQUATION AXB + CYD = E

On solutions to the quaternion matrix equation AXB+CYD=E

Expressions, as well as necessary and sufficient conditions are given for the existence of the real and pure imaginary solutions to the consistent quaternion matrix equation AXB+CY D = E. Formulas are established for the extreme ranks of real matrices Xi, Yi, i = 1, · · · , 4, in a solution pair X = X1 +X2i+X3j+X4k and Y = Y1+Y2i+Y3j+Y4k to this equation. Moreover, necessary and sufficient cond...

The submatrix constraint problem of matrix equation AXB+CYD=E

We say that X = [xij ]i,j=1 is symmetric centrosymmetric if xij = xji and xn−j+1,n−i+1, 1 ≤ i, j ≤ n. In this paper we present an efficient algorithm for minimizing ‖AXB + CY D − E‖ where ‖ · ‖ is the Frobenius norm, A ∈ Rt×n, B ∈ Rn×s, C ∈ Rt×m, D ∈ Rm×s, E ∈ Rt×s and X ∈ Rn×n is symmetric centrosymmetric with a specified central submatrix [xij ]r≤i,j≤n−r, Y ∈ Rm×m is symmetric with a specifie...

Diagonal and Monomial Solutions of the Matrix Equation AXB=C

In this article, we consider the matrix equation $AXB=C$, where A, B, C are given matrices and give new necessary and sufficient conditions for the existence of the diagonal solutions and monomial solutions to this equation. We also present a general form of such solutions. Moreover, we consider the least squares problem $min_X |C-AXB |_F$ where $X$ is a diagonal or monomial matrix. The explici...

An efficient iterative method for solving the matrix equation AXB + CYD = E

Iterative solutions to the linear matrix equation AXB + CXTD = E

In this paper the gradient based iterative algorithm is presented to solve the linear matrix equation AXB +CXD = E, where X is unknown matrix, A,B,C,D,E are the given constant matrices. It is proved that if the equation has a solution, then the unique minimum norm solution can be obtained by choosing a special kind of initial matrices. Two numerical examples show that the introduced iterative a...