The Structure and Distribution of Roots of 2×2 Matrices

Number Of Solutions ◽

Jordan Canonical Form ◽

The Matrix ◽

Explicit Formulae ◽

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.

Lorentz Boosts and Wigner Rotations: Self-Adjoint Complexified Quaternions

Physics ◽

10.3390/physics3020024 ◽

2021 ◽

Vol 3 (2) ◽

pp. 352-366

Author(s):

Thomas Berry ◽

Matt Visser

Keyword(s):

Point Of View ◽

Sufficient Condition ◽

Use Of Self ◽

Wigner Rotation ◽

Inertial Frames ◽

Explicit Formulae ◽

Specific Implementation ◽

Lorentz Boosts

In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated that, for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. The Wigner rotation is subsequently related to the generic non-associativity of the composition of three 4-velocities, and a necessary and sufficient condition is developed for the associativity to hold. Finally, the authors relate the composition of 4-velocities to a specific implementation of the Baker–Campbell–Hausdorff theorem. As compared to ordinary 4×4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae.

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

Journal of Mathematics ◽

10.1155/2021/6651434 ◽

2021 ◽

Vol 2021 ◽

pp. 1-14

Author(s):

Jin Wang

Keyword(s):

Matrix Equation ◽

Matrix Multiplication ◽

Sufficient Condition ◽

Mathematical Tool ◽

The Matrix ◽

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

Volume 1B: 25th Biennial Mechanisms Conference ◽

10.1115/detc98/mech-5878 ◽

1998 ◽

Author(s):

Namik Ciblak ◽

Harvey Lipkin

Keyword(s):

Recursive Algorithm ◽

Computer Applications ◽

Sufficient Condition ◽

Orthonormal Bases ◽

Isotropic Vector ◽

The Matrix ◽

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

Journal of Systems Science and Information ◽

10.21078/jssi-2018-459-14 ◽

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 ◽

The Matrix ◽

Product Approach ◽

Matrix Expression ◽

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

Journal of Algebra and Its Applications ◽

10.1142/s0219498821502169 ◽

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 ◽

The Matrix ◽

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

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100040329 ◽

1966 ◽

Vol 62 (4) ◽

pp. 673-677 ◽

Cited By ~ 22

Author(s):

M. H. Pearl

Keyword(s):

Generalized Inverse ◽

Sufficient Conditions ◽

Maximal Rank ◽

Sufficient Condition ◽

The Real ◽

Finite Dimensional ◽

The Matrix ◽

Generalized Inverses Of Matrices ◽

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

Canadian Mathematical Bulletin ◽

10.4153/cmb-1997-059-6 ◽

1997 ◽

Vol 40 (4) ◽

pp. 498-508

Author(s):

Chikkanna Selvaraj ◽

Suguna Selvaraj

Keyword(s):

Bounded Variation ◽

Matrix Transformations ◽

Sufficient Condition ◽

Summability Methods ◽

The Matrix ◽

Dirichlet Convolution ◽

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.

On Efficient Constructions of Lightweight MDS Matrices

IACR Transactions on Symmetric Cryptology ◽

10.46586/tosc.v2018.i1.180-200 ◽

2018 ◽

pp. 180-200

Author(s):

Lijing Zhou ◽

Licheng Wang ◽

Yiru Sun

Keyword(s):

Matrix Polynomial ◽

Hadamard Matrix ◽

Sufficient Condition ◽

Mds Matrix ◽

Residue Ring ◽

Maximum Distance Separable ◽

The Matrix ◽

Binary Matrices ◽

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.

Stability of Vlasov equilibria. Part 3. Models

Journal of Plasma Physics ◽

10.1017/s0022377800026362 ◽

1982 ◽

Vol 27 (1) ◽

pp. 37-53 ◽

Cited By ~ 21

Author(s):

Charles E. Seyler ◽

H. Ralph Lewis

Keyword(s):

Fluid Model ◽

Alternative Form ◽

Sufficient Condition ◽

Model Stability ◽

Explicit Formulae ◽

The Stability ◽

Vlasov Plasma

In this third of a series of papers on the stability of Vlasov equilibria, the dispersion functional technique developed in part 1 is applied to the multi-species Vlasov plasma and to the Vlasov-fluid model. An alternative form of the dispersion functional is derived for the Vlasov-fluid model in which magnetohydrodynamic and kinetic aspects of the problem are separated explicitly. A necessary and sufficient condition is derived for stability with the Vlasov-fluid model. Stability of the multi-species Vlasov plasma is discussed and a necessary and sufficient condition for stability near a marginal point is derived. Explicit formulae for quantities occurring in the dispersion functional are given in cylindrical co-ordinates for the Vlasov-fluid model when there is one nonignorable co-ordinate in the equilibrium. The dispersion functional for a multi-species Vlasov plasma is shown to be related to the ponderomotive Hamiltonian.

Representations and sign pattern of the group inverse for some block matrices

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.3169 ◽

2015 ◽

Vol 30 ◽

pp. 744-759

Author(s):

Lizhu Sun ◽

Wenzhe Wang ◽

Changjiang Bu ◽

Yimin Wei ◽

Baodong Zheng

Keyword(s):

Block Matrix ◽

Sufficient Condition ◽

Group Inverse ◽

Real Matrix ◽

Sign Pattern ◽

Block Matrices ◽

The Matrix ◽

Inverse Representation

Let $M= \left[ \begin{array}{cc} A& B \\ C& O \end{array} \right]$ be a complex square matrix where A is square. When BCB^{\Omega} =0, rank(BC) = rank(B) and the group inverse of $\left[ \begin{array}{cc} B^{\Omega} A B^{\Omega} & 0 \\ CB^{\Omega} & 0 \right]$ exists, the group inverse of M exists if and only if rank(BC + A)B^{\Omega}AB^{\Omega})^{\pi}B^{\Omega}A)= rank(B). In this case, a representation of $M^#$ in terms of the group inverse and Moore-Penrose inverse of its subblocks is given. Let A be a real matrix. The sign pattern of A is a (0,+,â)-matrix obtained from A by replacing each entry by its sign. The qualitative class of A is the set of the matrices with the same sign pattern as A, denoted by Q(A). The matrix A is called S^2GI, if the group inverse of each matrix \bar{A} in Q(A) exists and its sign pattern is independent of e A. By using the group inverse representation, a necessary and sufficient condition for a real block matrix to be an S^2GI-matrix is given.