On generalized inverses of matrices over integral domains
TLDR
In this article, it was shown that a matrix A admits a 1-inverse if and only if a linear combination of all the r × r minors of A is equal to one, where r is the rank of A.About:
This article is published in Linear Algebra and its Applications.The article was published on 1983-02-01 and is currently open access. It has received 46 citations till now. The article focuses on the topics: Matrix (mathematics) & Integral domain.read more
Citations
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Core–EP inverse
TL;DR: This work introduces a notion of ‘core–EP inverse’ for a square matrix which is not essentially of index one, and obtained a formula to compute the core–EP generalized inverse from a particular linear combination of minors of given matrix.
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Generalized inverses over integral domains
TL;DR: In this paper, it was shown that if the integral domain is a principal ideal domain, every generalized inverse can be obtained by that procedure, and it was also shown that a matrix A of rank r over an integral domain has Moore-Penrose inverse if and only if the sum of squares of all r × r minors of A is an invertible element of the integral domains.
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The generalized Moore-Penrose inverse
TL;DR: In this article, the generalized Moore-Penrose inverse was defined and sufficient conditions for its existence over an integral domain were given, and a generalized Cramer's rule was proposed.
Book
Linear algebra and linear models
TL;DR: In this article, the authors introduce vector spaces and matrices, linear estimation, tests of linear hypothesis, Singular values and their applications, block designs and optimization, and rank additivity.
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Full-rank and determinantal representation of the Drazin inverse
TL;DR: In this paper, a full-rank representation of the Drazin inverse A D of a given complex matrix A is introduced, which is based on an arbitrary full rank decomposition of A l, l⩾k, where k is the index of A.
References
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Book
Linear statistical inference and its applications
TL;DR: Algebra of Vectors and Matrices, Probability Theory, Tools and Techniques, and Continuous Probability Models.
Journal ArticleDOI
Linear Statistical Inference and its Applications
P. G. Moore,C. Radhakrishna Rao +1 more
TL;DR: The theory of least squares and analysis of variance has been studied in the literature for a long time, see as mentioned in this paper for a review of some of the most relevant works. But the main focus of this paper is on the analysis of variance.
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Linear Statistical Inference and Its Applications.
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On generalized inverses of polynomial and other matrices
TL;DR: In this paper, necessary and sufficient conditions are given for a rectangular multivariable polynomial matrix to have a (weak) generalized inverse, extending recent results for single-variable matrices obtained by other authors.