S.W. Drury

The Fatou-Zygmund property for Sidon sets

In the terminology of Edwards, Hewitt and Ross [2], the set X has the Fatou-Zygmund property. We refer the reader to this article and to Ross [7] for a deeper understanding of the content of Theorem 1. The proof of Theorem 1 uses the technique of [3] but the presentation we give is akin to that of [4]. Unexplained notations and definitions may be found in [5]. For technical reasons we should like X to be a finite set. Thus we shall actually prove the following result.

Principal powers of matrices with positive definite real part

We study principal powers of complex square matrices with positive definite real part, or with numerical range contained in a sector. We extend the notion of geometric mean to such matrices and we establish an operator norm bound in this context.

Sharp Condition Number Estimates for the Symmetric 2-Lagrange Multiplier Method

Domain decomposition methods are used to find the numerical solution of large boundary value problems in parallel. In optimized domain decomposition methods, one solves a Robin subproblem on each subdomain, where the Robin parameter a must be tuned (or optimized) for good performance. We show that the 2-Lagrange multiplier method can be analyzed using matrix analytical techniques and we produce sharp condition number estimates.

A counterexample to a question of Merikoski and Virtanen on the compounds of unitary matrices

Abstract We provide a counterexample to a question of Merikoski and Virtanen relating to the determinantal conjecture of Marcus and de Oliveira.

A Fischer-type determinantal inequality

We obtain a containment region for a Fischer-type determinantal ratio for matrices with given angular numerical range less than or equal to a right angle.

The maximum Perron roots of digraphs with some given parameters

Abstract We characterize the extremal digraphs which attain the maximum Perron root of digraphs with given arc connectivity and number of vertices. We also characterize the extremal digraphs which attain the maximum Perron root of digraphs given diameter and number of vertices.

A bound for the determinant of certain Hadamard products and for the determinant of the sum of two normal matrices

Abstract We establish an upper bound for the absolute value of the determinant of the Hadamard (elementwise) product of a unitary matrix and a general complex matrix. In the case that the general matrix is fixed and the unitary matrix is allowed to vary, this estimate is best possible. As a corollary, we obtain an upper bound for the absolute value of the determinant of the sum of two normal matrices with specified eigenvalues. This corollary supports the determinantal conjecture of Marcus and de Oliveira.

On symmetric functions of the eigenvalues of the sum of two Hermitian matrices

Let A and B be hermitian matrices with given eigenvalues (a1,…an) and (b1,…bn) respectively. Let (t1,…tn) be the eigenvalues of A + B. We establish that ∏j=1n(λ + tj) ϵ co ∏j=1n(λ + aj+bσ(j)); σ ϵ Sn where co denotes the convex hull in the space of polynomials and Sn denotes the permutation group on {1,…,n}.

The canonical correlations of a 2×2 block matrix with given eigenvalues

Abstract Let 1⩽k⩽n/2, and A= A 11 A 12 [−1pt]A 21 A 22 be an n×n positive definite matrix so that A11 is k×k. Suppose that A has given eigenvalues λ1⩾⋯⩾λn>0. The singular values σj(A11−1/2A12A22−1/2) (j=1,…,k) are known as the canonical correlations of the partitioned matrix A and have been extensively studied with regard to the inefficiency of the ordinary least squares method in statistics. The object of this paper is to provide proofs of some new inequalities for the canonical correlations in terms of λ1,…,λn.

Some remarks on a conjecture of Boyle and Handelman

Abstract Boyle and Handelman have conjectured that whenever A is an n × n nonnegative matrix with rank A ⩽ r and Perron root λ1, the inequality det(λI − tA) ⩽ λn−r(λr−λ1r) holds for all real numbers λ satisfying λ ⩾ λ1. We introduce an analogous conjecture involving nonnegative central (class) functions on the permutation group Sn. The analogue of the rank condition in this context is a condition on the support of the nonabelian Fourier transform of the central function. We are able to establish that both conjectures are true in case 2r ⩾ n.