LeRoy B. Beasley

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.

Linear transformations on matrices: The invariance of commuting pairs of matrices

Let L be a linear transformation on the set of all n×n matrices over an algebraically closed field of characteristic 0. It is shown that if AB=BA implies L(A)L(B)=L(B)L(A) and if either L is nonsingular or the implication in the hypothesis can also be reversed, then L is a sum of a scalar multiple of a similarity transformation and a linear functional times the identity transformation.

RANK INEQUALITIES OVER SEMIRINGS

Inequalities on the rank of the sum and the product of two matrices over semirings are surveyed. Preferences are given to the factor rank, row and column ranks, term rank, and zero-term rank of matrices over antinegative semirings. During the past century a lot of literature has been devoted to in- vestigations of semirings. Brie∞y, a semiring is essentially a ring where only the zero element is required to have an additive inverse. Therefore, all rings are also semirings. Moreover, among semirings there are such combinatorially interesting systems as the Boolean algebra of subsets of a flnite set(with addition being union and multiplication being intersec- tion), nonnegative integers and reals(with the usual arithmetic), fuzzy scalars(with fuzzy arithmetic), etc. Matrix theory over semirings is an object of much study in the last decades, see for example (9). In particu- lar, many authors have investigated various rank functions for matrices over semirings and their properties, see (1, 3, 6, 7, 8, 12) and references there in. There are classical inequalities for the rank function ‰ of sums and products of matrices over flelds, see, for example (10, 11): The rank-sum inequalities:

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.

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 that preserve pairs of matrices which satisfy extreme rank properties

Abstract A pair of m × n matrices ( A , B ) is called rank-sum-maximal if rank( A + B )=rank( A )+rank( B ), and rank-sum-minimal if rank( A + B )=|rank( A )−rank( B )|. We characterize the linear operators that preserve the set of rank-sum-minimal matrix pairs, and the linear operators that preserve the set of rank-sum-maximal matrix pairs over any field with at least min( m , n )+2 elements and of characteristic not 2.

Spaces of matrices of equal rank

Abstract Let Mm,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of Mm,n(F), all of whose nonzero elements have rank k, is said to be essentially decomposable if there exist nonsingular mXn matrices U and V respectively such that for any element A, UAV has the form UAV= A 1 A 2 A 3 0 where A1 is iX(k–i) for some i⩽k. Theorem: If K is a space of rank k matrices, then either K is essentially decomposable or dim K ⩽k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n⩾2k–1.

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.