Bit-Shun Tam

Maximizing spectral radius of unoriented Laplacian matrix over bicyclic graphs of a given order

For every integer n≥4, it is proved that there is a unique graph of order n which maximizes the spectral radius of the unoriented Laplacian matrix over all bicyclic graphs of order n, namely, the graph obtained from the cycle C 4 by first adding a chord and then attaching n − 4 pendant edges to one end of the chord.

A Note on Extreme Correlation Matrices

An

On the Jordan form of an irreducible matrix with eventually nonnegative powers

n\times n

CP rank of completely positive matrices of order 5

complex Hermitian or real symmetric matrix is a correlation matrix if it is positive semidefinite and all its diagonal entries equal one. The collection of all

On matrices whose numerical ranges have circular or weak circular symmetry

n\times n

Circularity of the numerical range

correlation matrices forms a compact convex set. The extreme points of this convex set are called extreme correlation matrices. In this note, elementary techniques are used to obtain a characterization of extreme correlation matrices and a canonical form for correlation matrices. Using these results, the authors deduce most of the existing results on this topic, simplify a construction of extreme correlation matrices proposed by Grone, Pierce, and Watkins, and derive an efficient algorithm for checking extreme correlation matrices.

The numerical range of a nonnegative matrix

Abstract In terms of the concept of a Frobenius collection of elementary Jordan blocks which we introduce, we characterize the collection of elementary Jordan blocks that appear in the Jordan form of an irreducible m -cyclic eventually nonnegative matrix whose m th power is permutationally similar to a direct sum of m eventually positive matrices.

A Simple proof of the Goldberg–Straus theorem on numerical radii

Abstract J.H. Drew et al. [Linear and Multilinear Algebra 37 (1994) 304] conjectured that for n ⩾4, the completely positive (CP) rank of every n × n completely positive matrix is at most [ n 2 /4]. In this paper we prove that the CP rank of a 5×5 completely positive matrix which has at least one zero entry is at most 6, thus providing new supporting evidence for the conjecture.

On the distinguished eigenvalues of a cone-preserving map

In [18] among other equivalent conditions, it is proved that a square complex matrix A is permutationally similar to a block-shift matrix if and only if for any complex matrix B with the same zero pattern as A, W(B), the numerical range of B, is a circular disk centered at the origin. In this paper, we add a long list of further new equivalent conditions. The corresponding result for the numerical range of a square complex matrix to be invariant under a rotation about the origin through an angle of 2π/m, where m⩾2 is a given positive integer, is also proved. Many interesting by-products are obtained. In particular, on the numerical range of a square nonnegative matrix A, the following unexpected results are established: (i) when the undirected graph of A is connected, if W(A) is a circular disk centered at the origin, then so is W(B), for any complex matrix B with the same zero pattern as A; (ii) when A is irreducible, if λ is an eigenvalue in the peripheral spectrum of A that lies on the boundary of W(A), then λ is a sharp point of W(A). We also obtain results on the numerical range of an irreducible square nonnegative matrix, which strengthen or clarify the work of Issos [9] and Nylen and Tam [14] on this topic. Open questions are posed at the end.

Cross-positive matrices revisited☆

Abstract An equivalent condition on a 3-square complex or a 4-square real upper triangular matrix is found for its numerical range to be a circular disk centered at the origin. Sufficient conditions for the circularity of the numerical range of certain sparse matrices are also given in terms of graphs.