Robert Reams

An inequality for nonnegative matrices and the inverse eigenvalue problem

We present two versions of the same inequality, relating the maximal diagonal entry of a nonnegative matrix to its eigenvalues. We demonstrate a matrix factorization of a companion matrix, which leads to a solution of the nonnegative inverse eigenvalue problem (denoted the nniep) for 4×4 matrices of trace zero, and we give some sufficient conditions for a solution to the nniep for 5×5 matrices of trace zero. We also give a necessary condition on the eigenvalues of a 5×5 trace zero nonnegative matrix in lower Hessenberg form. Finally, we give a brief discussion of the nniep in restricted cases.

Uniqueness of matrix square roots and an application

Abstract Let A∈M n ( C ) . Let σ(A) denote the spectrum of A , and F(A) the field of values of A . It is shown that if σ(A)∩(−∞,0]=∅ , then A has a unique square root B∈M n ( C ) with σ(B) in the open right (complex) half plane. This result and Lyapunovs theorem are then applied to prove that if F(A)∩(−∞,0]=∅ , then A has a unique square root with positive definite Hermitian part. We will also answer affirmatively an open question about the existence of a real square root B∈M n ( R ) for A∈M n ( R ) with F(A)∩(−∞,0]=∅ , where the field of values of B is in the open right half plane.

Hadamard inverses, square roots and products of almost semidefinite matrices

Abstract Let A = (aij) be an n × n symmetric matrix with all positive entries and just one positive eigenvalue. Bapat proved then that the Hadamard inverse of A, given by A° (−1) = ( 1 a ij ) is positive semidefinite. We show that if moreover A is invertible then A°(−1) is positive definite. We use this result to obtain a simple proof that with the same hypotheses on A, except that all the diagonal entries of A are zero, the Hadamard square root of A, given by A° 1 2 = (a ij 1 2 ) , has just one positive eigenvalue and is invertible. Finally, we show that if A is any positive semidefinite matrix and B is almost positive definite and invertible then A ○ B ⪰ (1/ e T B −1 e )A .

Constructing copositive matrices from interior matrices

Let A be an n by n symmetric matrix with real entries. Using the l1-norm for vectors and letting S + = {x ∈ R n |||x||1 =1 ,x≥ 0} ,t he matrixA is said to be interior if the quadratic form x T Ax achieves its minimum on S + in the interior. Necessary and sufficient conditions are provided for a matrix to be interior. A copositive matrix is referred to as being exceptional if it is not the sum of a positive semidefinite matrix and a nonnegative matrix. A method is provided for constructing exceptional copositive matrices by completing a partial copositive matrix that has certain specified overlapping copositive interior principal submatrices.

Methods for constructing distance matrices and the inverse eigenvalue problem

Abstract Let D 1 ∈ R k×k and D 2 ∈ R l×l be two distance matrices. We provide necessary conditions on Z∈ R k×l in order that D= D 1 Z Z T D 2 ∈ R n×n be a distance matrix. We then show that it is always possible to border an n×n distance matrix, with certain scalar multiples of its Perron eigenvector, to construct an (n+1)×(n+1) distance matrix. We also give necessary and sufficient conditions for two principal distance matrix blocks D 1 and D 2 be used to form a distance matrix as above, where Z is a scalar multiple of a rank one matrix, formed from their Perron eigenvectors. Finally, we solve the inverse eigenvalue problem for distance matrices in certain special cases, including n=3,4,5,6 , any n for which there exists a Hadamard matrix, and some other cases.

ISOMETRIC TIGHT FRAMES

A d × n matrix, n ≥ d, whose columns have equal length and whose rows are orthonormal is constructed. This is equivalent to finding an isometric tight frame of n vectors in R d (or C d ), or writing the d × d identity matrix I = dn=1 Pi ,w here thePi are rank 1 orthogonal projections. The simple inductive procedure given shows that there are many such isometric tight frames.

Scaling of symmetric matrices by positive diagonal congruence

We consider the problem of characterizing n-by-n real symmetric matrices A for which there is an n-by-n diagonal matrix D, with positive diagonal entries, so that DAD has row (and column) sums 1. Under certain conditions we provide necessary and sufficient conditions for the existence of a scaling for A, based upon both the positive definiteness of A on a cone lying in the nonnegative orthant and the semipositivity of A. This generalizes known results for strictly copositive matrices. Also given are (1) a condition sufficient for a unique scaling; (2) a characterization of those positive semidefinite matrices that are scalable; and (3) a new condition equivalent to strict copositivity, which we call total scalability. When A has positive entries, a simple iterative algorithm (different from Sinkhorns) is given to calculate the unique scaling.

Constructions of trace zero symmetric stochastic matrices for the inverse eigenvalue problem

In the special case of where the spectrum σ = {λ1 ,λ 2 ,λ 3, 0, 0 ,..., 0} has at most three nonzero eigenvalues λ1 ,λ 2 ,λ 3 with λ1 ≥ 0 ≥ λ2 ≥ λ3 ,a ndλ1 + λ2 + λ3 =0 , the inverse eigenvalue problem for symmetric stochastic n × n matrices is solved. Constructions are provided for the appropriate matrices where they are readily available. It is shown that when n is odd it is not possible to realize the spectrum σ with an n × n symmetric stochastic matrix when λ3 �nd 3 2n−3 > λ2 λ3 ≥ 0, andit is shown that this boundis best possible.

Determining protein structure using the distance geometry program APA

APA is a computer program, written in C, designed to determine the three-dimensional structure of proteins using distance geometry. We present the sampling and convergence properties of APA, as tested on bovine pancreatic trypsin inhibitor (BPTI). The results confirm the program’s earlier success with poly-L-alanine, albeit with some complications. The correct overall orientation of the BPTI conformation is achieved at an early stage in the algorithm. The correct orientations of the a-carbons are obtained by local reflections, instead of a penalty term, resulting in a smoother convergence. Finally, a process of choosing dissimilarities from two reduced data sets resulted in almost all structures converging. In order to compare with Havel’s DG-II distance geometry program, the sampling and convergence properties were tested on Havel’s 10 data sets. These simulated data sets were generated from the BPTI crystal and kindly provided by Tim Havel. # 1999 Elsevier Science Ltd. All rights reserved.

Integral similarity and commutators of integral matrices

Abstract Let F be a field, M n (F) the algebra of n × n matrices over F , and A ∈ M n (F) with trace( A )0. The following facts are well known: (i) if A is not a scalar, then A is similar over F to a matrix with zero diagonal; (ii) A [ P , Q ] PQ − QP for some P,Q ∈ M n (F) . We consider the situation when F is replaced by the ring of integers Z . We show that (ii) holds in this case for every n ⩾1. This result has been proved for n 2 by Lissner and Vaserstein (independently). We show also that (i) holds if F is replaced by Z for n >2 provided A ≢ aI mod p for all integers a and primes p .