On the tensor Permutation Matrices

We have expressed the tensor commutation matrix n ⊗ n as linear combination of the tensor products of the generalized Gell-Mann matrices. The tensor commutation matrices 3 ⊗ 2 and 2 ⊗ 3 have been expressed in terms of the classical Gell-Mann matrices and the Pauli matrices.

Linear Algebra and its Applications, 1991

Let P be an n X n permutation matrix, and let p be the corresponding permutation. Let A be a matrix such that AP = PA. It is well known that when p is an n-cycle, A is permutation similar to a circulant matrix. We present results for the band patterns in A and for the eigenstructure of A when p consists of several disjoint cycles. These results depend on the greatest common divisors of pairs of cycle lengths. (1991) Q

Journal of Mathematical Sciences, 2009

Elementary information on polynomials with tensor coefficients and operations with them is given. A generalized Bezout theorem is stated and proved, and on this basis, the Hamilton-Cayley theorem is proved. Another proof of the latter theorem is also considered. Several important theorems are proved, which apply in deducing of the formula expressing the adjunct tensor B (λ) for the tensor binomial λ (2p) E − A in terms of the tensor A ∈ C2p(Ω) (elements of this module are complex tensors of rank 2p) and its invariants. Furthermore, the definitions of minimal polynomial of the tensor of module C2p(Ω), of the tensor of module Cp(Ω) (whose elements are complex tensors of rank p), and of the tensor of module Cp(Ω) with respect to the given tensor of module C2p(Ω) are given. Here, Ω is some domain of the n-dimensional Euclidean (Riemannian) space. Some theorems concerning minimal polynomials are stated and proved. Moreover, the first, second, and third theorems on the splitting of the module Cp(Ω) into invariant submodules are given. Special attention is paid to theorems on adjoint, normal, Hermitian, and unitary tensors of modules C2p(Ω) and R2p(Ω) (elements of this module are real tensors of rank 2p). The theorem on polar decomposition [4, 6, 9, 13, 14], the Schur theorem [6], and the existence theorems for a general complete orthonormal system of eigentensors for a finite or infinite set of pairwise commuting normal tensors of modules C2p(Ω) and R2p(Ω) are generalized to tensors of a complex module of an arbitrary even order. Canonical representations of normal, conjugate, Hermitian, and unitary tensors of the module C2p(Ω) are given (the definition of this module can be found in [3, 17]). Moreover, the Cayley formulas for linear operators [6] are generalized to tensors of the module C2p(Ω). CONTENTS Definition 1.1. The tensor B(λ) ∈ C 2p (Ω), whose components are polynomials with respect to λ, is called a polynomial with tensor coefficients, or tensor polynomial , or λ-tensor. By virtue of the definition, the components of the tensor polynomial B(λ) ∈ C 2p (Ω) are represented in the form B i 1 .

arXiv: General Mathematics, 2018

This paper is concentrated on the classification of permutation matrix with the permutation similarity relation, mainly about the canonical form of a permutational similar equivalence class, the cycle matrix decomposition of a permutation matrix and the cycle factorization of a permutation matrix or monomial matrix.

2008

The partial transpose of a block matrix M is the matrix obtained by transposing the blocks of M independently. We approach the notion of the partial transpose from a combinatorial point of view. In this perspective, we solve some basic enumeration problems concerning the partial transpose of permutation matrices. More specifically, we count the number of permutations matrices which are invariant under the partial transpose and the number of permutation matrices whose partial transposes are still permutations. We solve these problems also when restricted to transposition matrices only.

International Journal of Contemporary Mathematical Sciences, 2006

Since tensors are vectors, no matter their order and valencies, they can be represented in matrix form and in particular as column matrices of components with respect to their corresponding bases. This consideration allows using all the available tools for vectors and can be extremely advantageous to deal with some interesting problems in linear algebra, as solving tensor linear equations. Once, the tensors have been operated as vectors, they can be returned to their initial notation as tensors. This technique, that is specially useful for implementing computer programs, is illustrated by several examples of applications. In particular, some interesting tensor and matrix equations are solved. Several numerical examples are used to illustrate the proposed methodology.

2016

In [Journal of Statistical Planning and Inference (141) (2011) 3697-3704], Roberto Fontana offers an algorithm for obtaining Sudoku matrices. Introduced by Geir Dahl concept disjoint pairs of S-permutation matrices [Linear Algebra and its Applications (430) (2009) 2457-2463] is used in this algorithm. Analyzing the works of G. Dahl and R. Fontana, the question of finding a general formula for counting disjoint pairs of n 2 × n 2 S-permutation matrices as a function of the integer n naturally arises. This is an interesting combinatorial problem that deserves its consideration. The present work solves this problem. To do that, the graph theory techniques have been used. It has been shown that to count the number of disjoint pairs of n 2 × n 2 S-permutation matrices, it is sufficient to obtain some numerical characteristics of the set of all bipartite graphs of the type g = R g ∪ C g , E g , where V = R g ∪ C g is the set of vertices, and E g is the set of edges of the graph g, R g ∩ C g = ∅, |R g | = |C g | = n.

ArXiv, 2012

The concept of S-permutation matrix is considered in this paper. When two binary matrices are disjoint is defined. For an arbitrary $n^2 \times n^2$ S-permutation matrix, the number of all disjoint whit it S-permutation matrices is found. A formula for counting the number of all disjoint pairs of $n^2 \times n^2$ S-permutation matrices is formulated and proven. In particular, a new shorter proof of a known equality is obtained in the work.

Zeitschrift für Operations Research, 1986

This note deals with the problem of permuting elements within columns of a real matrix so as to minimize a real-valued function of row sums. The special case dealing with minimization of maximum row sum has been studied by several authors 6, recently. Here we are concerned primarily with the case in which the matrix has two columns only and the function is Schur-convex.

Taiwanese Journal of Mathematics, 2020

We establish some necessary and sufficient conditions for the solvability to a system of a pair of coupled two-sided Sylvester-type tensor equations over the quaternion algebra. We also give an expression of the general solution to the system when it is solvable. As applications, we derive some solvability conditions and expressions of the η-Hermitian solutions to some systems of coupled two-sided Sylvester-type quaternion tensor equations. Moreover, we provide an example to illustrate the main results of this paper.