scholarly journals On the Necessary and Sufficient Condition for a Set of Matrices to Commute and Some Further Linked Results

2009 ◽  
Vol 2009 ◽  
pp. 1-24 ◽  
Author(s):  
M. De La Sen
Keyword(s):  
Null Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complex Matrices ◽  
Kronecker Sum ◽  
Structured Matrix ◽  
Commuting Matrices ◽  
The Matrix ◽  
Necessary And Sufficient

This paper investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator[A,B]=0for two matricesAandBif and only if a vectorv(B)defined uniquely from the matrixBis in the null space of a well-structured matrix defined as the Kronecker sumA⊕(−A∗), which is always rank defective. This result is extendable directly to any countable set of commuting matrices. Complementary results are derived concerning the commutators of certain matrices with functions of matricesf(A)which extend the well-known sufficiency-type commuting result[A,f(A)]=0.

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 On Solutions of the Matrix Equation A ∘ l X  = B with respect to M M -2 Semitensor Product

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Jin Wang
Keyword(s):  
Matrix Equation ◽  
Matrix Multiplication ◽  
Sufficient Condition ◽  
Mathematical Tool ◽  
Necessary And Sufficient Condition ◽  
The Matrix ◽  
Necessary And Sufficient

M M -2 semitensor product is a new and very useful mathematical tool, which breaks the limitation of traditional matrix multiplication on the dimension of matrices and has a wide application prospect. This article aims to investigate the solutions of the matrix equation A ° l X = B with respect to M M -2 semitensor product. The case where the solutions of the equation are vectors is discussed first. Compatible conditions of matrices and the necessary and sufficient condition for the solvability is studied successively. Furthermore, concrete methods of solving the equation are provided. Then, the case where the solutions of the equation are matrices is studied in a similar way. Finally, several examples are given to illustrate the efficiency of the results.

Orthonormal Isotropic Vector Bases

1998 ◽  
Author(s):  
Namik Ciblak ◽  
Harvey Lipkin
Keyword(s):  
Recursive Algorithm ◽  
Computer Applications ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Orthonormal Bases ◽  
Isotropic Vector ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Deviatoric Stresses ◽  
Isotropic Vectors

Abstract Orthonormal bases of isotropic vectors for indefinite square matrices are proposed and solved. A necessary and sufficient condition is that the matrix must have zero trace. A recursive algorithm is presented for computer applications. The isotropic vectors of 3 × 3 matrices are solved explicitly. Deviatoric stresses in continuum mechanics, the existence of isotropic vectors (particularly in screw space), and stiffness synthesis by springs are shown to be related to the isotropic vector problem.

Finding Cut-Edges and the Minimum Spanning Tree via Semi-Tensor Product Approach

2018 ◽  
Vol 6 (5) ◽  
pp. 459-472
Author(s):  
Xujiao Fan ◽  
Yong Xu ◽  
Xue Su ◽  
Jinhuan Wang
Keyword(s):  
Tensor Product ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Matrix ◽  
Product Approach ◽  
Matrix Expression ◽  
Necessary And Sufficient ◽  
Logical Functions

Abstract Using the semi-tensor product of matrices, this paper investigates cycles of graphs with application to cut-edges and the minimum spanning tree, and presents a number of new results and algorithms. Firstly, by defining a characteristic logical vector and using the matrix expression of logical functions, an algebraic description is obtained for cycles of graph, based on which a new necessary and sufficient condition is established to find all cycles for any graph. Secondly, using the necessary and sufficient condition of cycles, two algorithms are established to find all cut-edges and the minimum spanning tree, respectively. Finally, the study of an illustrative example shows that the results/algorithms presented in this paper are effective.

Matrix representation of formal polynomials over max-plus algebra

2020 ◽  
pp. 2150216
Author(s):  
Cailu Wang ◽  
Yuegang Tao
Keyword(s):  
Matrix Method ◽  
Polynomial Algorithm ◽  
Polynomial Function ◽  
Matrix Representation ◽  
Sufficient Condition ◽  
Polynomial Functions ◽  
Canonical Forms ◽  
Necessary And Sufficient Condition ◽  
The Matrix ◽  
Necessary And Sufficient

This paper proposes the matrix representation of formal polynomials over max-plus algebra and obtains the maximum and minimum canonical forms of a polynomial function by standardizing this representation into a canonical form. A necessary and sufficient condition for two formal polynomials corresponding to the same polynomial function is derived. Such a matrix method is constructive and intuitive, and leads to a polynomial algorithm for factorization of polynomial functions. Some illustrative examples are presented to demonstrate the results.

On generalized inverses of matrices

1966 ◽  
Vol 62 (4) ◽  
pp. 673-677 ◽  
Author(s):  
M. H. Pearl
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Maximal Rank ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Finite Dimensional ◽  
The Matrix ◽  
Generalized Inverses Of Matrices ◽  
Necessary And Sufficient

The notion of the inverse of a matrix with entries from the real or complex fields was generalized by Moore (6, 7) in 1920 to include all rectangular (finite dimensional) matrices. In 1951, Bjerhammar (2, 3) rediscovered the generalized inverse for rectangular matrices of maximal rank. In 1955, Penrose (8, 9) independently rediscovered the generalized inverse for arbitrary real or complex rectangular matrices. Recently, Arghiriade (1) has given a set of necessary and sufficient conditions that a matrix commute with its generalized inverse. These conditions involve the existence of certain submatrices and can be expressed using the notion of EPr matrices introduced in 1950 by Schwerdtfeger (10). The main purpose of this paper is to prove the following theorem:Theorem 2. A necessary and sufficient condition that the generalized inverse of the matrix A (denoted by A+) commute with A is that A+ can be expressed as a polynomial in A with scalar coefficients.

Matrix Transformations Based on Dirichlet Convolution

1997 ◽  
Vol 40 (4) ◽  
pp. 498-508
Author(s):  
Chikkanna Selvaraj ◽  
Suguna Selvaraj
Keyword(s):  
Bounded Variation ◽  
Matrix Transformations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Summability Methods ◽  
The Matrix ◽  
Dirichlet Convolution ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractThis paper is a study of summability methods that are based on Dirichlet convolution. If f(n) is a function on positive integers and x is a sequence such that then x is said to be Af-summable to L. The necessary and sufficient condition for the matrix Af to preserve bounded variation of sequences is established. Also, the matrix Af is investigated as ℓ − ℓ and G − G mappings. The strength of the Af-matrix is also discussed.

The Structure and Distribution of Roots of 2×2 Matrices

2013 ◽  
Vol 765-767 ◽  
pp. 667-669
Author(s):  
Yuan Yuan Li
Keyword(s):  
Infinitely Many Solutions ◽  
Sufficient Condition ◽  
Singular Matrix ◽  
Necessary And Sufficient Condition ◽  
Number Of Solutions ◽  
Jordan Canonical Form ◽  
The Matrix ◽  
Explicit Formulae ◽  
Necessary And Sufficient ◽  
Distribution Of Roots

This paper is concerned with Jordan canonical form theorem of algebraic formulae giving all the solutions of the matrix equation Xm= A where n is a positive integer greater than 2 and A is a 2 × 2 matrix with real or complex elements. If A is a 2 × 2 non-singular matrix, the equation Xm = A has infinitely many solutions and we obtain explicit formulae giving all the solutions. If A is a 2 × 2 singular matrix, and we obtained necessary and sufficient condition of square root . This leads to very simple formulae for all the solutions when A is either a singular matrix or a non-singular matrix with two coincident eigenvalues. We also determine the precise number of solutions in various cases.

Characterisation of the factors of quasi-differential expressions

1992 ◽  
Vol 120 (3-4) ◽  
pp. 297-312
Author(s):  
D. Race ◽  
A. Zettl
Keyword(s):  
Differential Expression ◽  
Null Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
General Scalar ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition for a general, scalar, quasi-differential expression of order n to be factorisable into a product of expressions of order n − k and k, for any 0 < k < n, is given. The factors are characterised completely in terms of elements of the null space of the expression and its adjoint. The results obtained extend existing results due to both Polya and Zettl from the case of classical linear differential expressions to quasi-differential expressions.

scholarly journals On Efficient Constructions of Lightweight MDS Matrices

2018 ◽  
pp. 180-200
Author(s):  
Lijing Zhou ◽  
Licheng Wang ◽  
Yiru Sun
Keyword(s):  
Matrix Polynomial ◽  
Hadamard Matrix ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Mds Matrix ◽  
Residue Ring ◽  
Maximum Distance Separable ◽  
The Matrix ◽  
Binary Matrices ◽  
Necessary And Sufficient

The paper investigates the maximum distance separable (MDS) matrix over the matrix polynomial residue ring. Firstly, by analyzing the minimal polynomials of binary matrices with 1 XOR count and element-matrices with few XOR counts, we present an efficient method for constructing MDS matrices with as few XOR counts as possible. Comparing with previous constructions, our corresponding constructions only cost 1 minute 27 seconds to 7 minutes, while previous constructions cost 3 days to 4 weeks. Secondly, we discuss the existence of several types of involutory MDS matrices and propose an efficient necessary-and-sufficient condition for identifying a Hadamard matrix being involutory. According to the condition, each involutory Hadamard matrix over a polynomial residue ring can be accurately and efficiently searched. Furthermore, we devise an efficient algorithm for constructing involutory Hadamard MDS matrices with as few XOR counts as possible. We obtain many new involutory Hadamard MDS matrices with much fewer XOR counts than optimal results reported before.

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