Best Approximate Solution of Matrix Equation AXB+CYD=E

Authors:
An-Ping Liao;Zhong-Zhi Bai;Yuan Lei
Affiliations:
-;-;-
Venue:
SIAM Journal on Matrix Analysis and Applications
Year:
2005

Citing 0
Cited 3

An iterative method for the symmetric and skew symmetric solutions of a linear matrix equation AXB+CYD=E

Journal of Computational and Applied Mathematics
Nested splitting conjugate gradient method for matrix equation AXB=C and preconditioning

Computers & Mathematics with Applications
The reflexive least squares solutions of the matrix equation $A_1X_1B_1+A_2X_2B_2+\cdots +A_lX_lB_l=C$with a submatrix constraint

Numerical Algorithms

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Abstract

We derive an analytical expression of the best approximate solution in the least-squares solution set ${\mathbb S}_E$ of the matrix equation $AXB+CYD=E$ to a given matrix pair $(X_f, Y_f)$, where $A$, $B$, $C$, $D$, and $E$ are given matrices of suitable sizes. Our work is based on the projection theorem in the finite-dimensional inner product space, and we use the generalized singular value decomposition and the canonical correlation decomposition. Moreover, we establish a direct method for computing this best approximate solution. An algorithm for finding the best approximate solution is described in detail, and an example is used to show the feasibility and effectiveness of our algorithm.