scholarly journals Rank Function and Outer Inverses

2018 ◽  
Vol 33 ◽  
pp. 16-23
Author(s):  
Manjunatha Prasad Karantha ◽  
K. Nayan Bhat ◽  
Nupur Nandini Mishra
Keyword(s):  
Commutative Ring ◽  
Linear Equations ◽  
Rank Function ◽  
Drazin Inverse ◽  
Rank Condition ◽  
Outer Inverse ◽  
The Matrix ◽  
Class Of Matrices ◽  
Necessary And Sufficient ◽  
Row Space

For the class of matrices over a field, the notion of `rank of a matrix' as defined by `the dimension of subspace generated by columns of that matrix' is folklore and cannot be generalized to the class of matrices over an arbitrary commutative ring. The `determinantal rank' defined by the size of largest submatrix having nonzero determinant, which is same as the column rank of given matrix when the commutative ring under consideration is a field, was considered to be the best alternative for the `rank' in the class of matrices over a commutative ring. Even this determinantal rank and the McCoy rank are not so efficient in describing several characteristics of matrices like in the case of discussing solvability of linear system. In the present article, the `rank--function' associated with the matrix as defined in [{\it Solvability of linear equations and rank--function}, K. Manjunatha Prasad, \url{http://dx.doi.org/10.1080/00927879708825854}] is discussed and the same is used to provide a necessary and sufficient condition for the existence of an outer inverse with specific column space and row space. Also, a rank condition is presented for the existence of Drazin inverse, as a special case of an outer inverse, and an iterative procedure to verify the same in terms of sum of principal minors of the given square matrix over a commutative ring is discussed.

scholarly journals A Dice Rolling Game on a Set of Tori

10.37236/2082 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Jeehoon Kang ◽  
Suh-Ryung Kim ◽  
Boram Park
Keyword(s):  
Commutative Ring ◽  
Linear Algebra ◽  
Main Idea ◽  
Additional Term ◽  
Combinatorial Problems ◽  
Algebraic Combinatorics ◽  
Combinatorial Structure ◽  
Constructive Proof ◽  
The Matrix ◽  
Row Space

Some linear algebraic and combinatorial problems are widely studied in connection with $\sigma$-games. One particularissue is to characterize whether or not a given vector lies in the submodule generated by the rows of a given matrix over a commutative ring. In general, one can solve this problem easily and algorithmically using the linear algebra over commutative ring. However, if the matrix has some combinatorial structure, one may expect that some more can be asserted instead of merely giving an algorithm. A recent outstanding example appeared in this line of research is the paper by Florence and Meunier published in Journal of Algebraic Combinatorics in 2010. In the same spirit, we consider a matrix over  $\mathbb{Z}_n$ to completely characterize the submodule generated by its rows and give a constructive proof. The main idea for the characterization is to find certain good basic elements in the row space and then express a given element as the linear combination of them as well as some additional term.

scholarly journals The outer inverse f(2)T,S of a homomorphism of right R-modules

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6459-6468
Author(s):  
Zhou Wang
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Outer Inverse ◽  
Basic Properties ◽  
Definition Of ◽  
Necessary And Sufficient

In this paper, we introduce the definition of the generalized inverse f(2)T,S, which is an outer inverse of the homomorphism f of right R-modules with prescribed image T and kernel S. Some basic properties of the generalized inverse f(2)T,S are presented. It is shown that the Drazin inverse, the group inverse and the Moore-Penrose inverse, if they exist, are all the generalized inverse f 2) T,S. In addition, we give necessary and sufficient conditions for the existence of the generalized inverse f(1,2)T,S.

scholarly journals Parametrized solutions $X$ of the system $AXA = AY A$ and $A^k Y AX = XAY A^k$

2019 ◽  
Vol 35 ◽  
pp. 503-510 ◽  
Author(s):  
David Ferreyra ◽  
Marina Lattanzi ◽  
Fabián Levis ◽  
Néstor Thome
Keyword(s):  
General Solution ◽  
Matrix Equation ◽  
Equation System ◽  
Drazin Inverse ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complex Matrices ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Parametrized Solutions

Let A and E be n × n given complex matrices. This paper provides a necessary and sufficient condition for the solvability to the matrix equation system given by AXA = AEA and AkEAX = XAEAk, for k being the index of A. In addition, its general solution is derived in terms of a G-Drazin inverse of A. As consequences, new representations are obtained for the set of all G-Drazin inverses; some interesting applications are also derived to show the importance of the obtained formulas.

scholarly journals Solusi Sistem Persamaan Linear Fuzzy

Matematika ◽  
2017 ◽  
Vol 16 (2) ◽  
Author(s):  
I. Irmawati ◽  
Icih Sukarsih ◽  
R. Respitawulan
Keyword(s):  
Fuzzy Numbers ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Real Coefficient ◽  
Necessary And Sufficient Condition ◽  
Unknown Variable ◽  
Fuzzy Variables ◽  
System A ◽  
The Matrix ◽  
Necessary And Sufficient

Abstract. Let A with A is n x n real coefficient matrix which is real numbers,  is vector of n unknown fuzzy variables, and   is  n fuzzy  constants vector. This system is named fuzzy linear equations system. To find  the solution of fuzzy linear equations system A , this system must be transformed into  with B is   2n x 2n coefficient matrix,  is 2n x 1 matrix of unknown variable , and  is 2n x 1 matrix of constants. The solution of   indirectly is the solution of A  because the matrix  corresponded to  is not necessarily fuzzy numbers. The necessary and sufficient condition to make the matrix  become the solution of A   is must be non negative. To help finding the solution  fuzzy linear equations system, on algorithm is built and implemented on Matlab.Keywords: Fuzzy Linear Equations System, Fuzzy  Numbers, Algorithm. Abstrak. Diberikan  A  dengan A adalah matriks koefisien berukuran n x n yang merupakan bilangan real,  adalah n variabel fuzzy yang tidak diketahui,  adalah vektor konstanta fuzzy dengan panjang n. Sistem tersebut dinamakan sistem persamaan linear fuzzy. Dalam mencari solusi sistem persamaan linear fuzzy A sistem tersebut harus ditransformasikan dalam bentuk  dengan B adalah matriks koefisien berukuran 2n x 2n, adalah matriks 2n x 1 dari variabel yang tidak diketahui, dan adalah matriks 2n x 1 dari konstanta. Solusi dari  tidak langsung menjadi solusi A , karena  yang bersesuaian dengan belum tentu berupa bilangan fuzzy. Syarat perlu dan cukup agar  merupakan solusi A  yaitu  harus non negatif. Untuk memudahkan mencari solusi dari sistem persamaan linear fuzzy perlu dibangun algoritma solusi sistem persamaan linear fuzzy dan implementasinya menggunakan Matlab.Kata Kunci : Sistem Persamaan Linear Fuzzy, Bilangan Fuzzy, Algoritma.

scholarly journals The outer generalized inverse of an even-order tensor

2020 ◽  
Vol 36 (36) ◽  
pp. 599-615
Author(s):  
Jun Ji ◽  
Yimin Wei
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Full Rank ◽  
Outer Inverse ◽  
The Matrix ◽  
Even Order ◽  
Rank Factorization ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

Necessary and sufficient conditions for the existence of the outer inverse of a tensor with the Einstein product are studied. This generalized inverse of a tensor unifies several generalized inverses of tensors introduced recently in the literature, including the weighted Moore-Penrose, the Moore-Penrose, and the Drazin inverses. The outer inverse of a tensor is expressed through the matrix unfolding of a tensor and the tensor folding. This expression is used to find a characterization of the outer inverse through group inverses, establish the behavior of outer inverse under a small perturbation, and show the existence of a full rank factorization of a tensor and obtain the expression of the outer inverse using full rank factorization. The tensor reverse rule of the weighted Moore-Penrose and Moore-Penrose inverses is examined and equivalent conditions are also developed.

On Some Properties of Semi-rings

2018 ◽  
pp. 35-50
Author(s):  
A. I. Belousov
Keyword(s):  
Linear Equations ◽  
Oriented Graph ◽  
Theoretical Computer Science ◽  
Alternative Methods ◽  
Main Role ◽  
Cost Matrix ◽  
Sequential Elimination ◽  
The Matrix ◽  
Systems Of Linear Equations ◽  
The Cost

The main objective of this paper is to prove a theorem according to which a method of successive elimination of unknowns in the solution of systems of linear equations in the semi-rings with iteration gives the really smallest solution of the system. The proof is based on the graph interpretation of the system and establishes a relationship between the method of sequential elimination of unknowns and the method for calculating a cost matrix of a labeled oriented graph using the method of sequential calculation of cost matrices following the paths of increasing ranks. Along with that, and in terms of preparing for the proof of the main theorem, we consider the following important properties of the closed semi-rings and semi-rings with iteration.We prove the properties of an infinite sum (a supremum of the sequence in natural ordering of an idempotent semi-ring). In particular, the proof of the continuity of the addition operation is much simpler than in the known issues, which is the basis for the well-known algorithm for solving a linear equation in a semi-ring with iteration.Next, we prove a theorem on the closeness of semi-rings with iteration with respect to solutions of the systems of linear equations. We also give a detailed proof of the theorem of the cost matrix of an oriented graph labeled above a semi-ring as an iteration of the matrix of arc labels.The concept of an automaton over a semi-ring is introduced, which, unlike the usual labeled oriented graph, has a distinguished "final" vertex with a zero out-degree.All of the foregoing provides a basis for the proof of the main theorem, in which the concept of an automaton over a semi-ring plays the main role.The article's results are scientifically and methodologically valuable. The proposed proof of the main theorem allows us to relate two alternative methods for calculating the cost matrix of a labeled oriented graph, and the proposed proofs of already known statements can be useful in presenting the elements of the theory of semi-rings that plays an important role in mathematical studies of students majoring in software technologies and theoretical computer science.

Spectral analysis of M/G/1 and G/M/1 type Markov chains

1996 ◽  
Vol 28 (01) ◽  
pp. 114-165 ◽  
Author(s):  
H. R. Gail ◽  
S. L. Hantler ◽  
B. A. Taylor
Keyword(s):  
Markov Chains ◽  
Generating Functions ◽  
Complex Analysis ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Equilibrium Analysis ◽  
Transform Method ◽  
Technical Problems ◽  
The Matrix ◽  
Transient Case

When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

scholarly journals Complex-analytic and matrix-analytic solutions for a queueing system with group service controlled by arrivals

2000 ◽  
Vol 13 (4) ◽  
pp. 415-427
Author(s):  
Lev Abolnikov ◽  
Alexander Dukhovny
Keyword(s):  
Block Size ◽  
Iteration Process ◽  
Analytic Solutions ◽  
Analytic Method ◽  
Service Time Distribution ◽  
Matrix Analytic Method ◽  
Queueing Process ◽  
The Matrix ◽  
Matrix Iteration ◽  
Necessary And Sufficient

A bulk M/G/1 system is considered that responds to large increases (decreases) of the queue during the service act by alternating between two service modes. The switching rule is based on two “up” and “down” thresholds for total arrivals over the service act. A necessary and sufficient condition for the ergodicity of a Markov chain embedded into the main queueing process is found. Both complex-analytic and matrix-analytic solutions are obtained for the steady-state distribution. Under the assumption of the same service time distribution in both modes, a combined complex-matrix-analytic method is introduced. The technique of “matrix unfolding” is used, which reduces the problem to a matrix iteration process with the block size much smaller than in the direct application of the matrix-analytic method.

scholarly journals On circulant codes with prescribed distances

1977 ◽  
Vol 16 (3) ◽  
pp. 361-369
Author(s):  
M. Deza ◽  
Peter Eades
Keyword(s):  
Sufficient Conditions ◽  
Difference Sets ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Weighing Matrices ◽  
Cyclic Difference Sets ◽  
The Matrix ◽  
Circulant Codes ◽  
Necessary And Sufficient ◽  
Square Matrix

Necessary and sufficient conditions are given for a square matrix to te the matrix of distances of a circulant code. These conditions are used to obtain some inequalities for cyclic difference sets, and a necessary condition for the existence of circulant weighing matrices.

scholarly journals Controllability and Observability of a Large Scale Thermodynamical System via Connectability Approach

2010 ◽  
Author(s):  
Virdiansyah Permana ◽  
Rahmat Shoureshi
Keyword(s):  
Graph Theory ◽  
Lie Algebra ◽  
Large Scale ◽  
Point Of View ◽  
Rank Condition ◽  
Computational Point ◽  
New Approach ◽  
Class System ◽  
Necessary And Sufficient ◽  
Controllability And Observability

This study presents a new approach to determine the controllability and observability of a large scale nonlinear dynamic thermal system using graph-theory. The novelty of this method is in adapting graph theory for nonlinear class and establishing a graphic condition that describes the necessary and sufficient terms for a nonlinear class system to be controllable and observable, which equivalents to the analytical method of Lie algebra rank condition. The directed graph (digraph) is utilized to model the system, and the rule of its adaptation in nonlinear class is defined. Subsequently, necessary and sufficient terms to achieve controllability and observability condition are investigated through the structural property of a digraph called connectability. It will be shown that the connectability condition between input and states, as well as output and states of a nonlinear system are equivalent to Lie-algebra rank condition (LARC). This approach has been proven to be easier from a computational point of view and is thus found to be useful when dealing with a large system.

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