Several Problems Of Rank-Constrained Matrix Equations

The contents of this thesis are divided into three parts.1. Extremal ranks of rank-constrained matrix expressions and rank-constrained least squares solutions of matrix equationsFirst, the minimal rank of the matrix expression A-BX with respect to r(X)=k are determined. Secondly, the minimal rank of the matrix expression A-BXBH with respect to r(X)=k, X≥0 are determined, where A∈Cm×m is a Hermitian matrix, and B∈Cm×n, r(B)=b. Finally, We derived the expression of least squares solutions to the matrix equation AXB=B, where A∈Cm×n, B∈Cp×q, C∈Cm×q be given, and X, r(C-AXB)= min, is a variant matrix.2. Several representations of generalized inversesFirst, we derived several representations of {2}-inverses of A∈Crm×n by using full-rank matrices A*,βand Aα,*. Secondly, some new representations of the Moore-Penrose inverse of A∈Crm×n are derived in terms of s×t-constrained submatrices with m≥s≥r,n≥t≥r. Finally, representations of the Moore-Penrose inverse and the group inverse of A-B were given3. Characterizations of EP matrices and weighted-EP matricesA complex square matrix A is said to be EP if A and its conjugate transpose AH have the same range. It is known result that A is EP if and only if r[A AH]=r(A). We first collect a group of known characterizations of EP matrix, and give some new characterizations of EP matrices. Then, we define weighted-EP matrix, and present a wealth of characterizations for weighted-EP matrix through various rank formulas for matrices and their generalized inverses.