AN INVERSE EIGENPROBLEM FOR GENERALIZED REFLEXIVE MATRICES WITH NORMAL {K+1}-POTENCIES

作者：Xu, Wei-Ru^*; Chen, Guo-Liang

来源：Electronic Journal of Linear Algebra, 2016, 31(1): 100-119.

DOI：10.13001/1081-3810.3173

• Left and right inverse eigenproblem
• Optimal approximation problem
• Generalized reflexive matrix
• Moore-Penrose generalized inverse
• {k+1}-Potent matrix

Let P, Q is an element of C-nxn be two normal {k + 1}-potent matrices, i.e., PP* = P*P, Pk+1 = P, QQ* = Q*Q, Q(k+1) = Q, k is an element of N. A matrix A is an element of C-nxn is referred to as generalized reflexive with two normal {k + 1}-potent matrices P and Q if and only if A = PAQ. The set of all n x n generalized reflexive matrices which rely on the matrices P and Q is denoted by gR(nxn) (P, Q). The left and right inverse eigenproblem of such matrices ask from us to find a matrix A is an element of gR(nxn) (P, Q) containing a given part of left and right eigenvalues and corresponding left and right eigenvectors. In this paper, first necessary and sufficient conditions such that the problem is solvable are obtained. A general representation of the solution is presented. Then an expression of the solution for the optimal Frobenius norm approximation problem is exploited. A stability analysis of the optimal approximate solution, which has scarcely been considered in existing literature, is also developed.

• 出版日期2016-3
• 单位华东师范大学