Thomas J Laffey | University College Dublin (original) (raw)

Papers by Thomas J Laffey

Linear Algebra and its Applications, Jul 1, 2020

Linear & Multilinear Algebra, Feb 1, 1990

ABSTRACT

Irish Mathematical Society Bulletin, 1978

arXiv (Cornell University), Jan 23, 2015

Let M n (F) be the algebra of n × n matrices and let S be a generating set of M n (F) as an F-alg... more Let M n (F) be the algebra of n × n matrices and let S be a generating set of M n (F) as an F-algebra. The length of a finite generating set S of M n (F) is the smallest number k such that words of length not greater than k generate M n (F) as a vector space. Traditionally the identity matrix is assumed to be automatically included in all generating sets S and counted as a word of length 0. In this paper we discuss how the problem changes if this assumption is removed.

Linear Algebra and its Applications, Apr 1, 2018

Let M n (F) be the algebra of n ×n matrices over the field F and let S be a generating set of M n... more Let M n (F) be the algebra of n ×n matrices over the field F and let S be a generating set of M n (F) as an F-algebra. The length of a finite generating set S of M n (F) is the smallest number k such that words of length not greater than k generate M n (F) as a vector space. It is a long standing conjecture of Paz that the length of any generating set of M n (F) cannot exceed 2n − 2. We prove this conjecture under the assumption that the generating set S contains a nonderogatory matrix. In addition, we find linear bounds for the length of generating sets that include a matrix with some conditions on its Jordan canonical form. Finally, we investigate cases when the length 2n − 2 is achieved.

arXiv (Cornell University), May 1, 2020

We consider the Karpelevič region Θ n ⊂ C consisting of all eigenvalues of all stochastic matrice... more We consider the Karpelevič region Θ n ⊂ C consisting of all eigenvalues of all stochastic matrices of order n. We provide an alternative characterisation of Θ n that sharpens the original description given by Karpelevič. In particular, for each θ ∈ [0, 2π), we identify the point on the boundary of Θ n with argument θ. We further prove that if n ∈ N with n ≥ 2, and t ∈ Θ n , then t is a subdominant eigenvalue of some stochastic matrix of order n.

arXiv (Cornell University), May 17, 2018

We show that if a list of nonzero complex numbers σ = (λ 1 , λ 2 ,. .. , λ k) is the nonzero spec... more We show that if a list of nonzero complex numbers σ = (λ 1 , λ 2 ,. .. , λ k) is the nonzero spectrum of a diagonalizable nonnegative matrix, then σ is the nonzero spectrum of a diagonalizable nonnegative matrix of order k + k 2 .

arXiv (Cornell University), Sep 27, 2019

Euclidean distance matrices corresponding to an arithmetic progression have rich spectral and str... more Euclidean distance matrices corresponding to an arithmetic progression have rich spectral and structural properties. We exploit those properties to develop completely positive factorizations of translations of those matrices. We show that the minimal translation that makes such a matrix positive semidefinite results in a completely positive matrix. We also discuss completely positive factorizations of such matrices over the integers. Methods developed in the paper can be used to find completely positive factorizations of other matrices with similar properties.

arXiv (Cornell University), Jan 25, 2017

In this paper we prove that the SNIEP = DNIEP, i.e. the symmetric and diagonalizable nonnegative ... more In this paper we prove that the SNIEP = DNIEP, i.e. the symmetric and diagonalizable nonnegative inverse eigenvalue problems are different. We also show that the minimum t > 0 for which (3 + t, 3 − t, −2, −2, −2) is realizable by a diagonalizable matrix is t = 1, and we distinguish diagonalizably realziable lists from general realizable lists using the Jordan Normal Form.

Banach Center Publications, 1994

In this survey paper, we present (mainly without proof) a number of results on conjugacy and fact... more 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

Irish Mathematical Society Bulletin, 1979

The American Mathematical Monthly, 1988

Linear Algebra and its Applications, 1980

Let A, B be n X n matrices with entries in an algebraically closed field F of characteristic zero... more Let A, B be n X n matrices with entries in an algebraically closed field F of characteristic zero, and let CAB -BA. It is shown tbat if C has rank two and A 'Bkk is nilpotent for 0 < i, i < n-1, 1 < k < 2, then A, B are simultaneously triangularizable over F. An example is given to show that this result is in some sense best possible. Let A,B be n X n matrices with entries in an algebraically closed field F. A well-known result of McCoy states that there exists a nonsingular matrix T over F such that T-'AT, T-'BT are both (upper) triangular if and only if p(A, B)(AB-BA) is nilpotent for all polynomials p(x, y) in the noncommuting indeterminates x, y over F. Let C = AB-BA. If C = 0, this condition is clearly satisfied and we have shown in [4, Theorem (1.4)] that it is also satisfied if C has rank one and F has characteristic zero or greater than n. This restriction on the characteristic has been removed by Guralnick [2]. However, the general problem of replacing McCoy's conditions by a "small" finite set of conditions is in general unsolved. See [4] for references to the known results. Unlike the case of C of rank one, it is possible to generate the full matrix algebra M,(F) of n X n matrices over F by a pair A, B with C of rank two, (see Gaines [l] am' also [4, (l.S)]). In this paper, we examine the case of C of rank two. The case where C2-0 is the most difficult case to deal with. It seems likely that in solving the general problem, the case C2=0 will also pose major difficulties. Our result. is

Linear Algebra and its Applications, 1985

Linear Algebra and its Applications, 1990

A characterization is given of all nonsingular linear operators, on the set of m X n matrices ove... more A characterization is given of all nonsingular linear operators, on the set of m X n matrices over any field with at least four elements, which map the set of rank-k matrices into itself. It is also shown that if 9 is any subspace of m X n matrices over any field with at least k f 1 elements whose nonzero elements all have rank k, then the dimension of 9 is at most max(m,n). This fact is used to characterize all linear operators on the set of m X n matrices over certain fields which map the set of rank-k matrices into itself. Let km,+@) denote the set of all m X n matrices over the field IF, and let p(A) denote the rank of A. We define a rank-k-preserver to be a linear operator T on d',.(F) such that p(A) = k implies p(T(A)) = k. Further, if T is a rank-k-preserver, we say that T preserves rank-k matrices. A linear operator which is a rank-k-preserver for each k = 1,2,.. . , min(m, n), is called a rank preserver.

Linear Algebra and its Applications, 1983

Let A, B be n X n matrices over a field F, and suppose A, B have quadratic minimal polynomials. T... more Let A, B be n X n matrices over a field F, and suppose A, B have quadratic minimal polynomials. Then the algebra generated by A, B has dimension at most 2n-1 if n is odd and 2n if n is even. These hounds are exact. In earlier work [l, p. 551, one of us (H.M.S.) showed that if A, B are n X n matrices over a field F of characteristic * 2 and A, B have quadratic minimal polynomials, then the algebra @ generated by A, B has dimension at most 2n. In this note we generalize this to arbitrary fields, showing in particular that for odd n, the bound on the dimension can be reduced to 2n-1. The methods are elementary.

Linear Algebra and its Applications, 1992

Let F be a field, and M,,(F) the algebra of n x n matrices over F. It is in general a very diffic... more Let F be a field, and M,,(F) the algebra of n x n matrices over F. It is in general a very difficult and tedious problem to determine the structure of the subalgebra X of M,(F) generated by a given subset S of M,(F). We show that if X contains a matrix A with n distinct eigenvalues in F, then X can be determined up to similarity very quickly by a graph-theoretic method. As a consequence we show that any such X can be generated by a pair of elements one of which can be taken to be A and the other a (0,l) matrix. As an application, we obtain a Specht-type similarity theorem. Let S be a nonempty set of n x n matrices. Define the directed graph G(S) of S as follows: G(S) has vertices 1,2,. .. , n; vertices i, j are joined by a directed edge if and only if there exists B = (b,,) E S with bij # 0. [Thus G(S) is the graph of the "generating matrix" CBEsxBB, where the xs are distinct indeterminates, in the usual sense.

Let A be an n × n (entrywise) positive matrix and let f (t) = det(I − tA). We prove the surprisin... more Let A be an n × n (entrywise) positive matrix and let f (t) = det(I − tA). We prove the surprising result that there always exists a positive integer N such that the formal power series expansion of 1 − f (t) 1/N around t = 0 has positive coefficients.

A constructive version of the celebrated Boyle-Handelman theorem on the non-zero spectra of nonne... more A constructive version of the celebrated Boyle-Handelman theorem on the non-zero spectra of nonnegative matrices is presented.