Equivalence Classes Of Matrices Research Articles

±1-Matrices with Vanishing Permanent

The problem of finding (−1, 1)-matrices with vanishing permanent was posed by Edward Wang in 1974. This paper states and proves bounds on the number of negative entries in a matrix with zero permanent and minimal number of negative entries among all matrices of the same equivalence class. Then representatives of every equivalence class of matrices with zero permanent are found for n ≤ 5. Bibliography: 20 titles.

An algorithmic version of the theorem by Latimer and MacDuffee for 2×2 integral matrices

Given two n× n integral matrices A and B, they are said to be equivalent if B= S −1 AS, where S is an n× n integral matrix with determinant ±1. If we consider n× n integral matrices with a fixed characteristic polynomial that is irreducible over Q , it is well known from a result by Latimer and MacDuffee that the number of matrix classes (equivalence classes of matrices) is equal to the number of ideal classes ( I ≅ J if I= qJ for some q in the quotient field) of the ring obtained by adjoining a root of the characteristic polynomial to Z . In this paper, we develop an effective version of this result for 2×2 matrices. We present an algorithm which given a 2×2 matrix finds a canonical representative in its class. In particular this allows us to determine whether two matrices are equivalent.

Procrustes analysis on some special manifolds

We pnwnt some ilicorciical results obtained by applying Procrustes methods to the statistical analysis on the two special manifolds, the Stiefel manifold Vk,m and the Grassmann manifold Gk,m−k or, equivalently, the manifold Pk,m−k of all m × m orthogonal projection matrices idempoteut of rank k. Procrustes representations of Vk,m and Pk,m−k by means of equivalence classes of matrices are considered, and Procrustes statistics and means are defined via the ordinary, weighted and generalized Procrustes methods. We discuss perturbation theory in Procrustes analysis on Vk,m and Pk,m−k. Finally, we give a brief discussion of embeddings of the Stiefel and Grassmann manifolds.

A dynamical system connected with the inhomogeneous six-vertex model. II. Evolution of orthogonal and symplectic matrices: An algebraic-geometric description

The present paper continues partI (Zap. Nauchn. Semin. POMI,215,178–196 (1994)), in which a dynamical system in discrete time generazed by a birational transformation acting on the equivalence classes of matrices of specific form is introduced. We study reductions of this system on orthogonal and symmetric matrices by means of algebraic geometry. Bibliography:4 titles.

A note of equivalence classes of matrices over a finite field

Let denote the algebra of m × m matrices over the finite field Fq of q elements, and let Ω denote a group of permutations of Fq. It is well known that each ϕϵΩ can be represented uniquely by a polynomial ϕ(x)ϵFq[x] of degree less than q; thus, the group Ω naturally determines a relation ∼ on as follows: if then A ~ B if ϕ(A) = B for some ϕϵΩ. Here ϕ(A) is to be interpreted as substitution into the unique polynomial of degree <q which represents ϕ.In an earlier paper by the second author [1], it is assumed that the relation ∼ is an equivalence relation and, based on this assumption, various properties of the relation are derived. However, if m ≥ 2, the relation ∼ is not an equivalence relation on . It is the purpose of this paper to point out the above erroneous assumption, and to discuss two ways in which hypotheses of the earlier paper can be modified so that the results derived there are valid.

Equivalence classes of matrices over a finite field

Let Fq = GF(q) denote the finite field of order q and F(m, q) the ring of m × m matrices over Fq. Let Ω be a group of permutations of Fq. If A, BϵF(m, q) then A is equivalent to B relative to Ω if there exists ϕϵΩ such that ϕ(A) = B where ϕ(A) is computed by substitution. Formulas are given for the number of equivalence classes of a given order and for the total number of classes induced by a cyclic group of permutations.

The Kronecker product of graphs

Introduction. This note considers a product derived from the Kronecker product of matrices. Some indication of the geometrical nature of this product is given and a theorem stating necessary and sufficient conditions for a product to be connected is proved. The matrix analogue of the above result is also stated. I. A convenient representation for a finite undirected [1] G is an adjacency matrix. If the vertex set of G is { Pi }, i = 1, , then an adjacency matrix of G is an nXn matrix (aij) with ai1 equal to the number of lines (paths of length one) joining pi to pj. A given is then represented by an equivalence class of matrices A= { PiAPi-l for all permutation matrices Pi of order equal to the order of A }. Each element of A: corresponds to a different ordering of the vertices of G. It is clear that for each such class of adjacency matrices there corresponds a unique class of isomorphic graphs. From this point on graph will mean a finite undirected with no loops. Such a has an adjacency matrix (ai0) whose entries are non-negative integers such that aij=aji and ai=0. We also use the following notation: o(G) is the number of vertices of G and is called the order of G, pLp7, is a chain in G from vertex Pi to vertex Pk, and n(pl->pk) is the number of lines (not necessarily distinct) in PlPk. If A and B are two adjacency matrices and A o B is some matrix product which is also an adjacency matrix then this matrix operation