Thomas J. Laffey

The real and the symmetric nonnegative inverse eigenvalue problems are different

We show that there exist real numbers Al, A2 ... A, that occur as the eigenvalues of an entry-wise nonnegative n-by-n matrix but do not occur as the eigenvalues of a symmetric nonnegative n-by-n matrix. This solves a problem posed by Boyle and Handelman, Hershkowitz, and others. In the process, recent work by Boyle and Handelman that solves the nonnegative inverse eigenvalue problem by appending 0s to given spectral data is refined.

Products of Matrices

We consider the problem of expressing an element A in GL(n, F), where F is a given field, as a product of elements in certain given distinguished subsets of GL(n,F). In particular, we consider the following types of decomposition: (i) A as a multiplicative commutator X -1 Y -1 XY (assuming detA = ±1). (ii) A as a product of involutions (assuming detA = ±1). (iii) A as a product of two involutions (assuming A is similar to A -1). (iv) A as a product of a symmetric matrix by an involution. (v) A as a product of skew-symmetric matrices.

A characterization of trace zero nonnegative 5×5 matrices

Let σ=(λ1,λ2,λ3,λ4,λ5) be a list of complex numbers with λ1+λ2+λ3+λ4+λ5=0. Necessary and sufficient conditions for the existence of an entry-wise nonnegative 5×5 matrix A with spectrum σ are presented.

Products of idempotent matrices

A result of J. Erdos [2] states that if A is a singularn×n matrix with entries in a field F then A can be written as the product of idempotents over F. C. S. Ballantinc III quantified this result by relating the minimum number of idempotents required to the rank of A and in particular proved that A can be written as the product of n idempotents over F. In this paper we consider the case where F is replaced by a ring. We show thai if R is a division ring or a Euclidean ring, then every singular n×n matrix with entries in R can be expressed as a product of idempotents over R.

Simultaneous reduction of sets of matrices under similarity

Abstract Let S be a set of n × n matrices over a field F, and A 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.

The minimal dimension of maximal commutative subalgebras of full matrix algebras

Abstract Let F be a field and A a maximal commutative subalgebra of the full matrix algebra Mn(F). It is shown that dim A > (2n) 2 3 − 1. It is also shown that if the radical of A has cube zero, then dim A ⩾ [3n 2 3 − 4], and that this result is best possible for infinitely many natural numbers n.

Algebras generated by two idempotents

Abstract It is shown that a noncommutative simple algebra generated over a field F by two idempotents is necessarily the ring of 2×2 matrices over a simple extension of F, and that every matrix ring over a field K can be generated over K by three idempotents.

EXTREME NONNEGATIVE MATRICES

Abstract An (entrywise) nonnegative n × n matrix A is extreme if its spectrum (λ1,…, λn) has the property that for all ϵ > 0, (λ1 − ϵ,…, λn − ϵ) is not the spectrum of a nonnegative matrix. It is proved that if A is an extreme nonnegative matrix, then there exists a nonzero nonnegative matrix Y such that AY = YA and AT ∘ Y = 0 where ∘ is the Hadamard (or entrywise) product and T denotes transpose.

Linear transformations on that preserve the Ky Fan k-norm and a remarkable special case when (n k) = (4, 2)

Given an n × n real matrix A. the Ky Fan k-norm of it is defined as the sum of its k largest singular values. We determine the structures of the linear transformations T on that preserve the Ky Fan k-norm. We show that except for (n,k) = (4,2), there exist orthogonal matrices U and V such that where A+ either denotes A or denotes At . For the particular case (n,k) = (4, 2)T is either of the usual form mentioned above or equivalent to the composition of the usual form with a particular operator that has many remarkable properties.

Two-generated commutative matrix subalgebras

Abstract Let F be a field, and let Mn(F) be the algebra of n×n matrices over F. Let A, B ∈ Mn(F) with AB = BA, and let A be the algebra generated by A, B over F.A theorem of Gerstenhaber [Ann. Math. 73:324–348 (1991)] states that the dimension of A is at most n. Gerstenhabers proof uses the methods of algebraic geometry. In this paper, we obtain a purely matrix-theoretic proof of the result. We also examine the case when equality occurs. The case where F is algebraically closed and A is indecomposable (under similarity) holds the key to the general situation, and in the indecomposable case, we obtain a Cayley-Hamilton-like theorem expressing Bk as a polynomial in I,B,…, Bk−1 with coefficients in F[A], where k denotes the number of blocks in the Jordan form of A. If all Jordan blocks of A have the same size, we obtain a nonderogatory-like condition on B which is equivalent to dim F A = n . We also show that in this case dim F A = n is equivalent to the maximality of A as a commutative subalgebra of Mn(F).