James R. Weaver

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.

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.

Real eigenvalues of nonnegative matrices which commute with a symmetric matrix involution

Abstract It is well known that if P is a nonnegative matrix, then its spectral radius is an eigenvalue of P (Perron-Frobenius theorem). In this paper it is shown that if P is an n × n nonnegative matrix and it commutes with a nonnegative symmetric involution when n =4 m +3, then (1) P has at least two real eigenvalues if n =4 m or 4 m + 2, (2) P has at least one real eigenvalue if n =4 m +1, and (3) P has at least three real eigenvalues if n =4 m +3, where m is a nonnegative integer and n ⩾ 1. Examples are given to show that these results are the best possible, and nonnegative symmetric involutions are classified.

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.

Diagonally Scaled Permutations and Circulant Matrices

Abstract If R is an n × n matrix over the complex field which is the product of a diagonal matrix D and a permutation matrix P, then R is called a diagonally scaled permutation matrix. We present the eigenstructure of R by observing that R is permutation similar to the direct sum of diagonally scaled permutation matrices of the form DC where D is a diagonal matrix and C is the circulant permutation. 0 1 0 0 ⋯ 0 0 0 1 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 0 ⋯ 1 1 0 0 0 ⋯ 0 The matrix DC is called a scaled circulant permutation matrix. We consider two cases for R = DC: when the scaling matrix D is nonsingular, and when D is singular. In the singular case R is nilpotent, and we are able to obtain upper and lower bounds on the index of nilpotency of R. We conclude with information about matrices that commute with a scaled permutation matrix. We are also able to represent an arbitrary n × n Toeplitz matrix as a sum of matrices of the form D(k,α,β)Ck for k = 1, …, n where D(k,α,β) is a diagonal matrix.

Properties of Tournaments Among Well-Matched Players

1. TOURNAMENTS. In an n-player round robin tournament, each player plays one match against each of the other n - 1 players. The win-loss outcomes of these matches can be conveniently recorded in a tournament matrix A = [aij] as follows: First label the players in any order as 1, 2,.. ., n. For each pair i and j, set ai1 = 1 if player i defeats player j, and set aij = 0 otherwise. If i j j, then exactly one of aij and aji is nonzero; when i = j, aii = 0. What properties of the matrix A are related to the strengths of the players? The simplest measure of strength is the number of matches that the player wins, and the row sums of A count the number of matches won by each player. We are interested in understanding tournaments among players who are well matched in the sense that each player wins about half of the matches played. If the number of players is odd, many properties of A are very well understood. If the number of players is even, however, the properties of A are far less well understood. Indeed, there are many easily stated questions that lead to hard, open problems. Some of these problems are the focus of this paper. We let I denote the identity matrix, we let J denote the square matrix all of whose entries are ones, and we let e denote the column vector whose entries are all ones. A matrix A whose entries are zeros and ones is a tournament matrix exactly when

Norms induced by symmetric guage functions

Let F n be the set of all n×1 column vectors over F, where F=R, A norm ∥ċ∥ on F n is permutationally invariant if and it is an absolute norm if A permutationally invariant absolute norm on F n is called a symmetric gauge function. Given a norm ∥ċ∥ on F n and a nonsingular matrix HeF n×n , one can define a norm ∥ċ∥H by The purpose of this note is to study the conditons on H for which the norm ∥ċ∥H is an absolute norm, a permutationally invariant norm, and a symmetric gauge function, respectively, if ∥ċ∥is a symmetric gauge function.

Row Sums and Inverse Row Sums for Nonnegative Matrices

For a nonnegative, irreducible matrix A, the grading of the row sums vector and the grading of the Perron vector are used to predict the grading of the row sums vector of

Fiedler Matrices and Their Factorization

( I - A )^{ - 1}

What are the linear isometries of 1p/n?

. This has applications to Markov chains.