Least Squares Solution of BXA^T=T over Symmetric, Skew-Symmetric, and Positive Semidefinite X

Abstract

Abstract. An efficient method based on the quotient singular value decomposition (QSVD) is used to solve the constrained least squares problem min T − BXA T F over symmetric, skewsymmetric, and positive semidefinite (maybe asymmetrical) X. The general expression of the solution is given and some necessary and sufficient conditions are derived about the solvability of the matrix equation BXA T = T . In each case, an algorithm is given for the unique solution when B and A are of full column rank.

Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations

“…In particular, when the matrices A, B and C are real, Chu [27] got the solvability conditions for the symmetric solutions of the linear matrix equation (1) by applying the generalized singular value decomposition (GSVD) of matrix, and Golub and Van Loan [28] and Yuan [29] obtained its optimal symmetric solution by an algorithmic approach. Moreover, in Reference [30], the symmetric, the skew-symmetric and the symmetric positive semidefinite solutions of the linear matrix equation (1) in the least-squares sense were derived. In general, when the matrices A, B and C are complex, Khatri and Mitra [31] discussed the consistency conditions for the linear matrix equation (1) and obtained the general expressions for its Hermitian or positive semidefinite solutions by using the generalized inverses of the involved matrices.…”

The least squares problem of the matrix equation A 1 X 1 B 1 T + A 2 X 2 B 2 T = T

“…when (1.1) is not consistent, are still interesting in the passed decades [1,3,5,8,10,13,14] . In literature, together with generalized inverse, matrix factorization techniques such as the well-known QR decomposition, singular value decomposition (SVD), generalized SVD (GSVD) [4,6,11,12] , canonical correlation decomposition (CCD) [7] , and quotient singular value decomposition (QSV-D) [2,9] , are widely used to simplify the matrix equations so that the resulting systems are diagonal with respect to new variables and they can be solved directly.…”

Iterative method for the least squares problem of a matrix equation with tridiagonal matrix constraint

“…Pan and Y. Lei positive semidefinite matrix constraints have been widely discussed in the literature [3,4,13,17,20]. However, we should point out that the preliminary estimation matrixX is derived by the finite element discretization method in finite element model updating problems, so the preliminary estimation matrixX usually possesses the sparse tridiagonal matrix structure [2].…”