Rank equalities related to outer inverses of matrices and applications

Abstract

Many thanks go to my thesis advisor George P. H. Styan for his guidance, invaluable discussions and thorough understanding during the preparation of this thesis.

Some matrix equations with applications†

“…Linear and Multilinear Algebra 1337 LEMMA 4.1 [29] Let A 2 C mÂn , B 1 2 C mÂq 1 , C 2 2 C p 1 Ân , B 3 2 C mÂq 2 , C 3 2 C p 2 Ân and pðX, Y, ZÞ…”

Some properties of submatrices in a solution to the matrix equation AXB=C with applications

“…The main reason for this is owing to the limitation of methods that are available for representing ranks of matrices. In recent years, the present author studied this topic and wrote a series of papers [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] on (extremal) ranks of matrices and their applications. It is believed that this work will serve as a new tool in matrix theory and its applications.…”

On reverse-order laws for least-squares g-inverses and minimum norm g-inverses of a matrix product

“…, AA − = BB − and so on, see [8][9][10][11][12][13][14][15][16]. As an extension of this paper, it may be of interest to investigate the reverse-order laws (ABC) ( …”

“…A simple rank formula for the difference of two outer inverses of a matrix is needed in the sequel. Lemma 1.3 (see [11]). Let X 1 and X 2 be a pair of outer inverses of a matrix A, that is,…”