Abstract:

Based on  the singular value decomposition (SVD) and quotient singular value decomposition (QSVD) of a matrix and matrix pair, the necessary and sufficient conditions were established for the existence of the symmetric solutions and the expressions for the inverse problem of a matrix  X^TAX=Bunder a submatrix constraint. Moreover, the optimal approximation problem to a given matrix in the solution set was considered, and the optimal approximation symmetric solution was obtained.

Key words: submatrix constraint, inverse problem, symmetric solutions, quotient singular value decomposition(QSVD), optimal approximation