Normal forms in matrix spaces

Banach Center Publications, 1994

In this survey paper, we present (mainly without proof) a number of results on conjugacy and factorization in general linear groups over fields and commutative rings. We also present the additive analogue in matrix rings of some of these results. The first section deals with the question of expressing elements in the commutator subgroup of the general linear group over a field as (simple) commutators. In Section 2, the same kind of problem is discussed for the general linear group over a commutative ring. In Section 3, the analogous question for additive commutators is discussed. The case of integer matrices is given special emphasis as this is an area of current interest. In Section 4, factorizations of an element A ∈ GL(n, F) (F a field) in which at least one of the factors preserves some form (e.g. is symmetric or skew-symmetric) is considered. An application to the size of abelian subgroups of finite p-groups is presented. In Section 5, a curious interplay between additive and multiplicative commutators in M n (F) (F a field) is identified for matrices of small size and a general factorization theorem for a polynomial using conjugates of its companion matrix is presented. Notation. The notation is standard. In particular, GL(n, R) denotes the group of invertible n×n matrices A such that A and A −1 have entries in the ring R and in the case R is commutative and has an identity, SL(n, R) denotes the subgroup of those elements A of GL(n, R) with det A = 1. A matrix A is called nonderogatory (or cyclic) if its minimal polynomial equals its characteristic polynomial. Equivalently A is nonderogatory if the only matrices which commute with A are the polynomials in A; cf. [G-L-R, pp. 299-300]. An involution is an element

Journal of Symbolic Computation, 2005

Let G be a finite group. It is easy to compute the character of G corresponding to a given complex representation, but much more difficult to compute a representation affording a given character. In part this is due to the fact that a class of equivalent representations contains no natural canonical representation. Although there is a large literature devoted to computing representations, and methods are known for particular classes of groups, we know of no general method which has been proposed which is practical for any but small groups. We shall describe an algorithm for computing an irreducible matrix representation R which affords a given character χ of a given group G. The algorithm uses properties of the structure of G which can be computed efficiently by a program such as GAP, theoretical results from representation theory, theorems from group theory (including the classification of finite simple groups), and linear algebra. All results in this paper have been implemented in the GAP package REPSN.

Journal of Algebra, 2007

Let F be an algebraically closed field of characteristic different from 2. Define the orthogonal group, On(F), as the group of n by n matrices X over F such that XX ′ = In, where X ′ is the transpose of X and In the identity matrix. We show that every nonsingular n by n skew-symmetric matrix over F is orthogonally similar to a bidiagonal skew-symmetric matrix. In the singular case one has to allow some 4-diagonal blocks as well. If further the characteristic is 0, we construct the normal form for the On(F)-similarity classes of skew-symmetric matrices. In this case, the known normal forms (as presented in the well known book by Gantmacher) are quite different. Finally we study some related varieties of matrices. We prove that the variety of normalized nilpotent n by n bidiagonal matrices for n = 2s + 1 is irreducible of dimension s. As a consequence the skew-symmetric nilpotent n by n bidiagonal matrices are shown to form a variety of pure dimension s.

Linear Algebra and its Applications, 2004

We show that if k is an algebraically closed field and G a not necessarily connected reductive linear algebraic group over k, then G(k) is solvable, nilpotent or abelian if and only if every finite subgroup of G(k) is solvable, nilpotent or abelian respectively. We also obtain the analogous result for compact subgroups of GL n (C).

Linear Algebra and its Applications, 1977

two complex vector spaces in terms of matrices.

Glasgow Mathematical Journal, 1991

In [5] we exhibited the construction of faithful irreducible matrix representations of p-groups E and constructed their extensions to a semidirect product E. H, in case E and H satisfied suitable conditions. One of the major conditions was that the prime p had to be odd.In this paper we assume the same conditions as in [5], but now with p = 2, in order to see if similar results can be obtained. Henceforth we will work with the following hypothesis.

Nonlinear Analysis-theory Methods & Applications, 1998

The aim of this paper is to demonstrate a specific application of Computer Algebra to bifurcation theory with symmetry. The classification of different bifurcation phenomena in case of several parameters is automated, based on a classification of Gröbner bases of possible tangent spaces. The computations are performed in new coordinates of fundamental invariants and fundamental equivariants, with the induced weighted ordering. In order to justify the approach the theory of intrinsic modules is applied. Results for the groups D 3 , Z 2 , and Z 2 × Z 2 demonstrate that the algorithm works independent of the group and that new results are obtained.

Linear Algebra and its Applications, 1986

Let S be a set of n x n matrices over a field F, and ~4 the algebra generated by S over F. The problem of deciding whether the elements of S can be simultaneously reduced (to block-triangular form with the diagonal blocks of some specified size) is considered, and an account is given of various methods used to attack the problem. Most of the techniques use representation theory to obtain information on A'. The problems of simultaneous triangularization, existence of common eigenvectors, etc. are also considered. The aim of the paper is to survey the methods used to attack these problems and to give some typical results. The paper does not contain many new results.

2 Vector spaces and linear transformations 5 2.1 Matrix representations of endomorphism . . . . . . . . . . . . 5 2.2 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.6 Inner products . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.7 Unitary transformations . . . . . . . . . . . . . . . . . . . . . 9

cmi.ac.in

Abstract. In this paper we describe the ring of invariants of the space of m-tuples of n×n matrices, under the action of SL(n)×SL(n) given by (A, B)·(X1,X2, ··· ,Xm) ↦→ (AX1Bt, AX2Bt, ··· , AXmBt). Determining the ring of invariants is the first step in the geometric approach to ...

Linear Algebra and Its Applications, 2008

• bilinear forms over an algebraically closed or real closed field;

1998

Abstract We describe an algorithm to determine whether a matrix group over a finite field, generated by a given set of matrices, contains one of the classical groups or the special linear group. The algorithm is based on finding elements with certain properties by independent random selection from the group. Very few matrix groups other than the classical groups contain subsets of elements with the required properties, and a classification of such matrix groups is the principal theoretical basis of the algorithm.

1982

Journal of the Australian Mathematical Society, 2008

Let G be isomorphic to a group H satisfying SL(d, q) ≤ H ≤ GL(d, q) and let W be an irreducible F q G-module of dimension at most d 2 . We present a Las Vegas polynomial-time algorithm which takes as input W and constructs a d-dimensional projective representation of G.