Weighted Reverse Order Law for the Weighted Moore-Penrose Inverse in Rings with Involution

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Dijana Mosić ◽  
Dragan S. Djordjević
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Reverse Order ◽  
Reverse Order Law ◽  
Rings With Involution ◽  
Necessary And Sufficient

Abstract We investigate some necessary and sufficient conditions for the reverse order law for the weighted Moore-Penrose inverse in rings with involution.

scholarly journals Some additive and multiplicative results for generalized inverses

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 2049-2057
Author(s):  
Jovana Nikolov-Radenkovic
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Generalized Inverses ◽  
Reverse Order ◽  
Reverse Order Law ◽  
Necessary And Sufficient ◽  
Regular Operators

In this paper we give necessary and sufficient conditions for A1{1,3} + A2{1, 3}+ ... + Ak{1,3} ? (A1 + A2 + ... + Ak){1,3} and A1{1,4} + A2{1,4} + ... + Ak{1,4} ? (A1 + A2 + ... + Ak){1,4} for regular operators on Hilbert space. We also consider similar inclusions for {1,2,3}- and {1,2,4}-i inverses. We give some new results concerning the reverse order law for reflexive generalized inverses.

Perturbation results and the forward order law for the Moore-Penrose inverse of a product

2018 ◽  
Vol 34 ◽  
pp. 514-525
Author(s):  
Nieves Castro-Gonzalez ◽  
Robert Hartwig
Keyword(s):  
Sufficient Conditions ◽  
Full Rank ◽  
Necessary And Sufficient Conditions ◽  
Reverse Order ◽  
Complex Matrices ◽  
Reverse Order Law ◽  
Necessary And Sufficient

New expressions are given for the Moore-Penrose inverse of a product $AB$ of two complex matrices. Furthermore, an expression for $(AB)\dg - B\dg A\dg$ for the case where $A$ or $B$ is of full rank is provided. Necessary and sufficient conditions for the forward order law for the Moore-Penrose inverse of a product to hold are established. The perturbation results presented in this paper are applied to characterize some mixed-typed reverse order laws for the Moore-Penrose inverse, as well as the reverse order law.

Further results on generalized inverses in rings with involution

2015 ◽  
Vol 30 ◽  
Author(s):  
N. Castro-Gonzalez ◽  
Jianlong Chen ◽  
Long Wang
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Generalized Inverses ◽  
Matrix Product ◽  
Explicit Formulas ◽  
Block Matrices ◽  
Unital Ring ◽  
Rings With Involution ◽  
Necessary And Sufficient

Let R be a unital ring with an involution. Necessary and sufficient conditions for the existence of the Bott-Duffin inverse of a in R relative to a pair of self-adjoint idempotents (e, f) are derived. The existence of a {1, 3}-inverse, {1, 4}-inverse, and the Moore-Penrose inverse of a matrix product is characterized, and explicit formulas for their computations are obtained. Some applications to block matrices over a ring are given.

scholarly journals On mixed-type reverse-order laws for the Moore-Penrose inverse of a matrix product

2004 ◽  
Vol 2004 (58) ◽  
pp. 3103-3116 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Mixed Type ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Reverse Order ◽  
Matrix Product ◽  
Matrix Rank ◽  
Matrix Rank Method ◽  
The Matrix ◽  
Necessary And Sufficient

Some mixed-type reverse-order laws for the Moore-Penrose inverse of a matrix product are established. Necessary and sufficient conditions for these laws to hold are found by the matrix rank method. Some applications and extensions of these reverse-order laws to the weighted Moore-Penrose inverse are also given.

scholarly journals Factorizations of Invertible Symmetric Matrices over Polynomial Rings with Involution

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Rings ◽  
Symmetric Matrices ◽  
Matrix Polynomials ◽  
Elementary Divisors ◽  
Rings With Involution ◽  
Ring Of Polynomials ◽  
Necessary And Sufficient

By means of the notions of infinite elementary divisors, dual and generalized dual matrix polynomials, we find necessary and sufficient conditions for the existence of factorizations of invertible symmetric matrices over ring of polynomials with involution.

scholarly journals The Moore-Penrose inverse in rings with involution

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 5791-5802 ◽  
Author(s):  
Sanzhang Xu ◽  
Jianlong Chen
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Unital Ring ◽  
Rings With Involution ◽  
Necessary And Sufficient

Let R be a unital ring with involution. In this paper, we first show that for an element a 2 R, a is Moore-Penrose invertible if and only if a is well-supported if and only if a is co-supported. Moreover, several new necessary and sufficient conditions for the existence of the Moore-Penrose inverse of an element in a ring R are obtained. In addition, the formulae of the Moore-Penrose inverse of an element in a ring are presented.

scholarly journals The mixed-type reverse order laws for generalized inverses of the product of two matrices

Filomat ◽  
2013 ◽  
Vol 27 (5) ◽  
pp. 937-947
Author(s):  
Zhiping Xiong
Keyword(s):  
Mixed Type ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Generalized Inverses ◽  
Reverse Order ◽  
Matrix Products ◽  
Schur Complements ◽  
Necessary And Sufficient ◽  
The Relationship

The relationship between generalized inverses of AB and the product of generalized inverses of A and B have been studied in this paper. The necessary and sufficient conditions for a number of mixed-type reverse order laws of generalized inverses of two matrix products are derived by using the maximal ranks of the generalized Schur complements.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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