Inverse problem for symmetric $P$-symmetric matrices with a submatrix constraint

Abstract

For a fixed generalized reflection matrix $P$, i.e., $P^T=P, P^2= I$, and $P\neq \pm I$, then a matrix $A $ is called a symmetric $P$-symmetric matrix if $A=A^T$ and $(PA)^T=PA$. This paper is mainly concerned with finding the least squares symmetric $P$-symmetric solutions to the matrix inverse problem $AX=B$ with a submatrix constraint, where $X$ and $B$ are given matrices of suitable size. By applying the generalized singular value decomposition and the canonical correlation decomposition, an analytical expression of the least squares solutions is derived basing on the Projection Theorem in Hilbert inner products spaces. Moreover, in the corresponding solution set, the analytical expression of the unique minimum-norm solution is described in detail.