An expression of the general common least-squares solution to a pair of matrix equations with applications

In this paper, an explicit representation of the general common least-squares solution to the pair of matrix equations A1XB1=C1A1XB1=C1 and A2XB2=C2A2XB2=C2 is obtained. Furthermore, we use this result to determine the condition for the existence of a Hermitian least-squares solution to the matrix equation AXB=CAXB=C, and the expression of the general Hermitian least-squares solution is also given. Special attention is paid to consider the existence of Hermitian {1,i}{1,i}-inverses of AA, i=3i=3, 4, and the representations of the Hermitian generalized inverses are presented. Finally, a numerical example is given to illustrate our result.