On integral \(EP_ r\) matrices
type: Item
instance of: scholarly article
title: On integral \(EP_ r\) matrices
zbMATH Open document ID: 0556.15010
DOI: 10.1007/BF01849020
author: Ar. Meenakshi
published in: Periodica Mathematica Hungarica
publication date: 1983-01-01T00:00:00Z
review text: A square complex matrix A is called \(EP_ r\) matrix, if \(rank(A)=r\) and \(R(A)=R(A^*)\) where \(A^*\) denotes the conjugate transpose of A and R(A) denotes the range of A. \textit{R. Penrose} [Proc. Camb. Philos. Soc. 51, 406-413 (1955; Zbl 0065.246)] has proved that there exists a unique matrix \(A^+\) that satisfies the following equations \(AA^+A=A\), \(A^+AA^+=A^+\), \((AA^+)=AA^+\), \((A^+A)=A^+A\). A matrix X is called an \(\{\) \(\ell \}\)-inverse of A if \(AXA=A.\) The main results of this paper are the following: 1. Let A be an \(n\times n\) integral matrix. Then the following statements are equivalent: (i) A is \(EP_ r\) and \(A^ TA\) has an integral \(\{\) \(\ell \}\)-inverse. (ii) There exists a permutation matrix P such that \(PAP^ T=\left[ \begin{matrix} D\quad 0\\ 0\quad 0\end{matrix} \right]\), where D is an \(r\times r\) nonsingular matrix with \(| \det D| =1\). (iii) \(A=GLG^ T\), where L and G are integral, \(G^ TG=I_ r\) and L is \(r\times r\) nonsingular with \(| \det L| =1\). (iv) \(A^+\) is an integral and \(EP_ r\) matrix. (v) There exists a symmetric idempotent integral matrix E such that \(AE=EA\) and \(R(A)=R(E).\) 2. Let A and B be integral \(EP_ r\) matrices. Then, \(A^ TA\) has an integral \(\{\) \(\ell \}\)-inverse and \(R(A)=R(B)\) if and only if AB is \(EP_ r\) and \((AB)^+=B^+A^+\) is integral.
reviewed by: K.Burian
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 3886064
zbMATH Keywords: integral \(EP_ r\) matrix
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: integral generalized inverses
MaRDI profile type: Publication
cites work: Integral generalized inverses of integral matrices
cites work: Further results on integral generalized inverses of integral matrices
cites work: On ranks of integral generalized inverses of integral matrices
cites work: More on integral generalised inverses of integral matrics
cites work: Theorems on products of EPr matrices. II
cites work: OnEPrmatrices with entries from an arbitrary field
cites work: Q5849795
cites work: Q3287260
Recommended article: Q3971610
Recommended article: Q3731774
Recommended article: Two sets of new characterizations for normal and EP matrices.
Recommended article: On the class of \(\{1,2,4\}\) inverses as solution of a linear problem
Recommended article: Idempotency of the Hermitian part of a complex matrix
Recommended article: A simple proof of the product theorem for EP matrices
Recommended article: Does the Generalized Inverse of A Commute with A?
Recommended article: Q4344250
Recommended article: Q4037939
Recommended article: Some remarks on generalized inverses of conjugate EP matrix
description: scientific article; zbMATH DE number 3886064