Jeffrey L. Stuart

Matrices that commute with a permutation matrix

Abstract Let P be an n × n permutation matrix, and let p be the corresponding permutation. Let A be a matrix such that AP = PA . It is well known that when p is an n -cycle, A is permutation similar to a circulant matrix. We present results for the band patterns in A and for the eigenstructure of A when p consists of several disjoint cycles. These results depend on the greatest common divisors of pairs of cycle lengths.

Irreducible Sign K-Potent Sign Pattern Matrices

Abstract The sign pattern matrix A is called sign k -potent if k is the smallest positive integer such that A k+1 =A. The structure of irreducible, sign k -potent pattern matrices was characterized by Stuart et al. (J. Stuart, C. Eschenbach, S. Kirkland, Linear Algebra Appl. 294 (1999) 85–92). We extend those results to the reducible case, providing necessary conditions that characterize the structure of each off-diagonal block of the Frobenius normal form of a reducible, sign k -potent matrix.

Reducible powerful ray pattern matrices

A ray pattern is a matrix each of whose entries is either 0 or a ray in the complex plane originating from 0 (but not including 0). A ray pattern is a natural generalization of the concept of a sign pattern, whose entries are from the set {+, −, 0}. Powers of sign patterns and ray patterns, especially patterns whose powers are periodic, have been studied in several recent papers. A ray pattern A is said to be powerful if Ak is unambiguously defined for all positive integers k. Irreducible powerful ray patterns have been characterized recently. In this paper, reducible powerful ray patterns are investigated. In particular, for a powerful ray pattern in Frobenius normal form, it is shown that the existence of a nonzero entry in an off diagonal block implies that the corresponding irreducible components are related in a certain way. Further, the structure of each of the off diagonal blocks is characterized.

On Almost Regular Tournament Matrices

Abstract Spectral and determinantal properties of a special class M n of 2n×2n almost regular tournament matrices are studied. In particular, the maximum Perron value of the matrices in this class is determined and shown to be achieved by the Brualdi–Li matrix, which has been conjectured to have the largest Perron value among all tournament matrices of even order. We also establish some determinantal inequalities for matrices in M n and describe the structure of their associated walk spaces.

Balancing environmental and industry sustainability: A case study of the US gold mining industry

Mandatory insurance requirements and/or mitigation fees (royalties) for mining companies may help reduce environmental risk exposure for the federal government. Mining is examined since the Environmental Protection Agency (EPA) Toxic Release Inventory reveals that this sector produces more hazardous waste than any other industrial sector. Although uncommon, environmental expense can exceed hundreds of millions of dollars per development. Of particular concern is the potential for mines to become unfunded Superfund sites. Monte Carlo simulation of risk exposure is used to establish a plausible range of unfunded federal liabilities associated with cyanide-leach gold mining. A model is developed to assess these costs and their impact on both the federal budget and corporate profitability (i.e., industry sustainability), particularly if such costs are borne by offending firms.

Properties of the Brualdi-Li tournament matrix

The Brualdi–Li tournament matrix is conjectured to have the largest spectral radius among all tournament matrices of even order. In this paper two forms of the characteristic polynomial of the Brualdi–Li tournament matrix are found. Using the first form it is shown that the roots of the characteristic polynomial are simple and that the Brualdi–Li tournament matrix is diagonalizable. Using the second form an expression is found for the coefficients of the powers of the variable λ in the characteristic polynomial. These coefficients give information about the cycle structure of the cycles of length 1–5 of the directed graph associated with the Brualdi–Li tournament matrix.

Irreducible, pattern k-potent ray pattern matrices

Abstract A ray pattern is the class of all complex matrices with a specified zero–nonzero pattern such that the argument of each nonzero entry is specified. The ray pattern A is called pattern k-potent if k is the smallest positive integer for which Ak+1=A as ray patterns. We characterize the irreducible ray patterns that are pattern k-potent, and we provide a canonical form for patterns with respect to permutation similarity and signature similarity by diagonal, unitary matrices.

Inflation matrices and ZME -matrices that commute with a permutation matrix

Centrosymmetric matrices are matrices that commute with the permutation matrix J, the matrix with ones on its cross-diagonal. This paper generalizes the concept of centrosymmetry, and considers the properties of matrices that commute with an arbitrary permutation matrix P, the P-commutative matrices. In particular,it focuses on two related classes of matrices: inflation matrices and

A polynomial time spectral decomposition test for certain classes of inverse M-matrices

ZME

Diagonally Scaled Permutations and Circulant Matrices

-matrices. The structure of P-commutative inflators is determined, and then this is used to characterize the P-commutative