Norman J. Pullman

Boolean-rank-preserving operators and Boolean-rank-1 spaces

Abstract We study the extent to which certain theorems on linear operators on field-valued matrices carry over to linear operators on Boolean matrices. We obtain analogues and near analogues of several such theorems. One of these leads us to consider linear spaces of m × n Boolean matrices whose nonzero members all have Boolean rank 1. We obtain a structure theorem for such spaces that enables us to determine the maximum Boolean dimension of such spaces and their maximum cardinality.

Semiring fank versus column rank

Abstract This paper concerns two notions of rank of matrices over semirings: semiring rank and column rank. These two rank functions are the same over fields and Euclidean rings, but differ for matrices over many combinatorially interesting semirings including the nonnegative integer matrices, the fuzzy matrices, and the Binary Boolean matrices. We investigate the largest value of r for which the column rank and semiring rank of all m × n matrices over a given semiring are both r . This value is determined for the semirings mentioned above as well as many others.

Term-rank, permanent, and rook-polynomial preservers

Characterizations are obtained of those linear operators on the m × n matrices over an arbitrary semiring that preserve term rank. We also present characterizations of permanent and rook-polynomial preserving operators on matrices over certain types of semirings. Our results apply to many combinatorially interesting algebraic systems, including nonnegative integer matrices, matrices over Boolean algebras, and fuzzy matrices.

Algebraic multiplicity of the eigenvalues of a tournament matrix

Let F” denote the set of irreducible n X n tournament matrices. Here arc our main results: (1) For all n > 3, every matrix in K has at least three distinct eigenvalues; such a matrix has exactly three distinct eigenvalues if and only if it is a Hadamard tournament matrix. (2) For all n 2 3 there is a matrix in Y” having n distinct eigenvalues. (3) If cr, denotes the maximum algebraic multiplicity of 0 as an eigenvalue of the matrices in z, then ln /21-2 6. (4) If r,, is the minimum Perron value (i.e. spectral radius) of all matrices in r”, then 2 8.

Tournament matrices and their generalizations, I.

If M is any complex matrix with rank (M + M * + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 for its real part. This generalizes a recent finding of de Caen and Hoffman: the rank of any n × n tournament matrix is at least n − 1. We extend several spectral properties of tournament matrices to this and related types of matrices. For example, we characterize the singular real matrices M with 0 diagonal for which rank (M + MT + I) = 1 and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible n × n tournament matrices exist if and only n∉ {2,3,4,5} and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented.

Nonnegative Rank-Preserving Operators

Abstract Analogues of characterizations of rank-preserving operators on field-valued matrices are determined for matrices witheentries in certain structures S contained in the nonnegative reals. For example, if S is the set of nonnegative members of a real unique factorization domain (e.g. the nonnegative reals or the nonnegative integers), M is the set of m×n matrices with entries in S , and min(m,n)⩾4, then a “linear” operator on M preserves the “rank” of each matrix in M if and only if it preserves the ranks of those matrices in M of ranks 1, 2, and 4. Notions of rank and linearity are defined analogously to the field-valued concepts. Other characterizations of rank-preserving operators for matrices over these and other structures S are also given.

Operators that preserve semiring matrix functions

Abstract Characterizations are obtained of those linear operators over certain semirings that preserve (1) the rth coefficient of the rook polynomial, those that preserve (2) the term rank of matrices with term rank r, and those that preserve (3) the rth elementary symmetric permanental function. Our results apply to many algebraic systems of combinatorial interest, including the nonnegative integer matrices, Boolean matrices, and fuzzy matrices.

Linear operators preserving idempotent matrices over fields

Abstract Suppose F is a field. We show that if the characteristic of the field is not 2, then the semigroup of linear operators on the n × n matrices over F that preserve idempotence is the group G ( F ) generated by transposition and similarity. Chan, Lim, and Tan have previously established that theorem for the real and complex fields by other methods. We also show that the semigroup L ( F ) of linear operators on the n × n matrices over F that preserve both idempotence and nonidempotence is G ( F ) when the characteristic of F is not 2. We determine the structure of L ( F ) when the characteristic of F is 2, and present some open problems.

Biclique coverings of regular bigraphs and minimum semiring ranks of regular matrices

We study the minimum number of complete bipartite subgraphs needed to cover and partition the edges of a k-regular bigraph on 2n vertices. Bounds are determined on the minima of these numbers for fixed n and k. Exact values of the minima are found for all n and k ≤ 4. The same results hold for directed graphs. Equivalently, we have determined bounds on the minimum value of the Boolean and nonnegative integer ranks of binary n × n matrices with constant row and column sum k for fixed n and k, obtaining the exact values of the minimum for k ≤ 4.

Fuzzy rank-preserving operators

Abstract Analogues of characterizations of rank-preserving operators on field-valued matrices are determined for fuzzy matrices and for matrices over related semirings. For example, if M is the set of mXn fuzzy matrices and min(m,n)>1, then a linear operator on M preserves the rank of each matrix in M if and only if it preserves the rank of those matrices in M of ranks 1 and 2. Other characterizations of rank-preserving operators are also given.