scholarly journals Group inverses and Drazin inverses of bidiagonal and triangular Toeqlitz matrices

1977 ◽  
Vol 24 (1) ◽  
pp. 10-34 ◽  
Author(s):  
R. E. Hartwig ◽  
J. Shoaf
Keyword(s):  
Toeplitz Matrices ◽  
Arbitrary Ring ◽  
Group Inverses ◽  
Necessary And Sufficient

AbstractNecessary and sufficient sonditions are given for the existence of the group and Drazin inverses of bidiagonal and triangular Toeplitz matrices over an arbitrary ring.

A note on the group inverses of block matrices over rings

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3685-3692
Author(s):  
Hanyu Zhang
Keyword(s):  
Associative Ring ◽  
Sufficient Conditions ◽  
Block Matrix ◽  
Necessary And Sufficient Conditions ◽  
Group Inverse ◽  
Block Matrices ◽  
Group Inverses ◽  
Necessary And Sufficient

Suppose R is an associative ring with identity 1. The purpose of this paper is to give some necessary and sufficient conditions for the existence and the representations of the group inverse of the block matrix (AX+YB B A 0) and M = (A B C D) under some conditions. Some examples are given to illustrate our results.

On the ergodicity of a class of level-dependent quasi-birth-and-death processes

2019 ◽  
Vol 51 (4) ◽  
pp. 1109-1128
Author(s):  
James D. Cordeiro ◽  
Jeffrey P. Kharoufeh ◽  
Mark E. Oxley
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Asymptotic Convergence ◽  
Birth And Death Processes ◽  
Positive Recurrence ◽  
Qbd Processes ◽  
Group Inverses ◽  
Necessary And Sufficient

AbstractWe examine necessary and sufficient conditions for recurrence and positive recurrence of a class of irreducible, level-dependent quasi-birth-and-death (LDQBD) processes with a block tridiagonal structure that exhibits asymptotic convergence in the rows as the level tends to infinity. These conditions are obtained by exploiting a multi-dimensional Lyapunov drift approach, along with the theory of generalized Markov group inverses. Additionally, we highlight analogies to well-known average drift results for level-independent quasi-birth-and-death (QBD) processes.

scholarly journals Explicit eigenvalues and inverses of several Toeplitz matrices

2006 ◽  
Vol 48 (1) ◽  
pp. 73-97 ◽  
Author(s):  
Wen-Chyuan Yueh ◽  
Sui Sun Cheng
Keyword(s):  
Difference Equations ◽  
Toeplitz Matrices ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Explicit Formulas ◽  
Necessary And Sufficient

AbstractBased on the theory of difference equations, we derive necessary and sufficient conditions for the existence of eigenvalues and inverses of Toeplitz matrices with five different diagonals. In the course of derivations, we are also able to derive computational formulas for the eigenvalues, eigenvectors and inverses of these matrices. A number of explicit formulas are computed for illustration and verification.

scholarly journals Toeplitz Matrices in the Problem of Semiscalar Equivalence of Second-Order Polynomial Matrices

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
B. Z. Shavarovskii
Keyword(s):  
Equivalence Relation ◽  
Polynomial Matrix ◽  
Toeplitz Matrices ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Matrices ◽  
Order Polynomial ◽  
Necessary And Sufficient

We consider the problem of determining whether two polynomial matrices can be transformed to one another by left multiplying with some nonsingular numerical matrix and right multiplying by some invertible polynomial matrix. Thus the equivalence relation arises. This equivalence relation is known as semiscalar equivalence. Large difficulties in this problem arise already for 2-by-2 matrices. In this paper the semiscalar equivalence of polynomial matrices of second order is investigated. In particular, necessary and sufficient conditions are found for two matrices of second order being semiscalarly equivalent. The main result is stated in terms of determinants of Toeplitz matrices.

On Group Inverses of Matrices with Order and Rank Perturbations

1994 ◽  
Vol 44 (3-4) ◽  
pp. 209-222 ◽  
Author(s):  
P.S.S.N.V.P. Rao
Keyword(s):  
Markov Chains ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Group Inverse ◽  
Necessary And Sufficient Condition ◽  
Matrix Formula ◽  
Group Inverses ◽  
Finite Markov Chains ◽  
Necessary And Sufficient

Group inverse of a square matrix A exists if and only if rank of A is equal to rank of A2. Group inverses have many applications, prominent among them is in the analysis of finite Markov chains discussed by Meyer (1982). In this note necessary and sufficient conditions for the existence of group inverses of bordered matrix, [Formula: see text] are obtained and expressions for the group inverses in terms of group inverse of A are given, whenever they exist. Also necessary and sufficient condition for the existence of group inverse of A in terms of group inverse of B and C are given. An application to perturbation in Markov chains is illustrated.

scholarly journals *-DMP elements in *-semigroups and *-rings

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3073-3085 ◽  
Author(s):  
Yuefeng Gao ◽  
Jianlong Chen ◽  
Yuanyuan Ke
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drazin Inverse ◽  
Complex Matrices ◽  
Arbitrary Ring ◽  
The Core ◽  
Core Inverse ◽  
Necessary And Sufficient

In this paper, we investigate *-DMP elements in *-semigroups and *-rings. The notion of *-DMP element was introduced by Patr?cio and Puystjens in 2004. An element a is *-DMP if there exists a positive integer m such that am is EP. We first characterize *-DMP elements in terms of the {1,3}-inverse, Drazin inverse and pseudo core inverse, respectively. Then, we characterize the core-EP decomposition utilizing the pseudo core inverse, which extends the core-EP decomposition introduced by Wang for complex matrices to an arbitrary *-ring; and this decomposition turns to be a useful tool to characterize *-DMP elements. Further, we extend Wang?s core-EP order from complex matrices to *-rings and use it to investigate *-DMP elements. Finally, we give necessary and sufficient conditions for two elements a,b in *-rings to have aaD = bbD, which contribute to study *-DMP elements.

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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