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Upper and lower bounds for ranks of matrix products composed by generalized inverses with applications

Yongge Tian(yongge.tian***at***gmail.com)
Bo Jiang(jiangboliumengyu***at***gmail.com)

Abstract: In matrix theory and its applications, we often meet with various matrix expressions or matrix equalities that involve inverses and generalized inverses of matrices, one of the simplest forms of such matrix expressions is given by $P^{(i,\ldots,j)}NQ^{(i,\ldots,j)}$, where $P$, $Q$, and $N$ are three given matrices, $P^{(i,\ldots,j)}$ and $Q^{(i,\ldots,j)}$ are the eight frequently used $\{i,\ldots, j\}$ generalized inverses of $P$ and $Q$. This paper presents a general approach by using the matrix rank methodology. We first establish exact formulas for calculating the maximum and minimum ranks of $P^{(i,\ldots,j)}NQ^{(i,\ldots,j)}$ with respect to $\{i,\ldots, j\}$-inverses of $P$ and $Q$, and then use the formulas to characterize a variety of algebraic properties of these products. Some applications to the special products $A^{(i,\ldots,j)}(A+B)B^{(i,\ldots,j)}$ and $(BC)^{(i,\ldots,j)}B(AB)^{(i,\ldots,j)}$ are also presented.

Keywords: 15A03; 15A09; 15A24

Category 1: Global Optimization


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Entry Submitted: 03/09/2017
Entry Accepted: 03/10/2017
Entry Last Modified: 03/09/2017

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