Linear operators that preserve maximal column ranks of nonnegative integer matrices

The maximal column rank of an by matrix over a semiring is the maximal number of the columns of which are linearly independent. We characterize the linear operators which preserve the maximal column ranks of nonnegative integer matrices.

     

Construction and nonnegativity of sign idempotent sign pattern matrices

In this paper, we modify Eschenbach’s algorithm for constructing sign idempotent sign pattern matrices so that it correctly constructs all of them. We find distinct classes of sign idempotent sign pattern matrices that are signature similar to an entrywise nonnegative sign pattern matrix. Additionally, if for a sign idempotent sign pattern matrix A there exists a signature matrix S such that SAS is nonnegative, we prove such S is unique up to multiplication by -1 if the signed digraph D(A) is not disconnected.     

三幂等符号模式矩阵的结构

Abstract. A matrix whose entries are , -, and 0 is called a sign pattern matrix. For a signpattern matrix A,if A3 =A, then A is said to be sign tripotent. In this paper, the characteriza-tion of the n by n(n≥2) sign pattern matrices A which are sign tripotent has been given out.Furthermore, the necessary and sufficient condition of A3=A but A2≠A is obtained, too.     

± sign pattern matrices that allow orthogonality

A sign pattern A is a ± sign pattern if A has no zero entries. A allows orthogonality if there exists a real orthogonal matrix B whose sign pattern equals A. Some sufficient conditions are given for a sign pattern matrix to allow orthogonality, and a complete characterization is given for ± sign patterns with n − 1 ⩽ N₋(A) ⩽ n + 1 to allow orthogonality.     

Primitive zero-symmetric sign pattern matrices with the maximum base

In [B. Cheng, B. Liu, The base sets of primitive zero-symmetric sign pattern matrices, Linear Algebra Appl. 428 (2008) 715-731], Cheng and Liu studied the bases of primitive zero-symmetric sign pattern matrices. The sharp upper bound of the bases was obtained. In this paper, we characterize the sign pattern matrices with the sharp bound.     

Primitive non-powerful sign pattern matrices with base 2

A sign pattern matrix M with zero trace is primitive non-powerful if for some positive integer k, M ^k ?=?J _#. The base l(M) of the primitive non-powerful matrix M is the smallest integer k. By considering the signed digraph S whose adjacent matrix is the primitive non-powerful matrix M, we will show that if l(M)?=?2, the minimum number of non-zero entries of M is 5n???8 or 5n???7 depending on whether n is even or odd.     

Imprimitive non-powerful sign pattern matrices with maximum base

You et al. [L.H. You, J.Y. Shao, and H.Y. Shan, Bounds on the bases of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007), pp. 285–300], obtained an upper bound of the bases for imprimitive non-powerful sign pattern matrices. In this article, we characterize those imprimitive non-powerful sign pattern matrices whose bases reach this upper bound.     

The nonnegative rank factorizations of nonnegative matrices

Let A ∈ P^{m × n}_r, the set of all m × n nonnegative matrices having the same rank r. For matrices A in P^{m × n}_n, we introduce the concepts of “A has only trivial nonnegative rank factorizations” and “A can have nontrivial nonnegative rank factorizations.” Correspondingly, the set P^{m × n}_n is divided into two disjoint subsets P₍₁₎ and P₍₂₎ such that P₍₁₎∪P₍₂₎ = P^{m × n}_n. It happens that the concept of “A has only trivial nonnegative rank factorizations” is a generalization of “A is prime in P^{n × n}_n.” We characterize the sets P₍₁₎ and P₍₂₎. Some of our results generalize some theorems in the paper of Daniel J. Richman and Hans Schneider [9].     

The base sets of primitive zero-symmetric sign pattern matrices

In [J.Y. Shao, L.H. You, Bound on the base of irreducible generalized sign pattern matrices, Discrete Math., in press], Shao and You extended the concept of the base from powerful sign pattern matrices to non-powerful (and generalized) sign pattern matrices. In this paper, we study the bases of primitive zero-symmetric sign pattern (and generalized sign pattern) matrices. Sharp upper bounds of the bases are obtained. We also show that there exist no “gaps” in the base sets of the classes of such matrices.     

On the Kapranov ranks of tropical matrices

We investigate the Kapranov rank functions of tropical matrices for different ground fields. For any infinite ground field we show that the rank-product inequality holds for Kapranov rank, and we prove that the Kapranov rank respects Green’s preorders on the semigroup of tropical n-by-n matrices. The rank-product inequality is shown to fail for Kapranov rank over any finite ground field. We provide an example of a 7-by-7 01-matrix whose Kapranov rank is independent of a ground field, equals 6, and exceeds tropical rank.     

On powers of nonnegative matrices

We investigate the structure of powers of nonnegative matrices A, and in particular characterize those A for which some power is (essentially) triangular. We also show how the number of irreducible components of A can be determined from its powers.     

Algebraic eigenspaces of nonnegative matrices

The Perron-Frobenius theory for square, irreducible, nonnegative matrices is generalized by studying the structure of the algebraic eigenspace of an arbitrary square nonnegative matrix corresponding to its spectral radius. We give a constructive proof that this subspace is spanned by a set of semipositive vectors and give a combinatorial characterization of both the index of the spectral radius and dimension of the algebraic eigenspace corresponding to the spectral radius. This involves a detailed study of the standard block triangular representation of nonnegative matrices by giving special attention to those blocks on the diagonal having the same spectral radius as the original matrix. We also show that the algebraic eigenspace corresponding to the spectral radius contains a semipositive vector having the largest set of positive coordinates among all vectors in this subspace.     

Intervals of totally nonnegative matrices

Totally nonnegative matrices, i.e., matrices having all their minors nonnegative, and matrix intervals with respect to the checkerboard ordering are considered. It is proven that if the two bound matrices of such a matrix interval are nonsingular and totally nonnegative (and in addition all their zero minors are identical) then all matrices from this interval are also nonsingular and totally nonnegative (with identical zero minors).     

On nonnegative factorization of matrices

It is shown that a sufficient condition for a nonnegative real symmetric matrix to be completely positive is that the matrix is diagonally dominant.     

A comparison of nonnegative real ranks and their preservers

This paper concerns three notions of rank of matrices over semirings; real rank, semiring rank and column rank. These three rank functions are the same over subfields of reals but differ for matrices over subsemirings of nonnegative reals. We investigate the largest values of r for which the real rank and semiring rank, real rank and column rank of all m×n matrices over a given semiring are both r, respectively. We also characterize the linear operators which preserve the column rank of matrices over certain subsemirings of the nonnegative reals.     

A comparison of nonnegative real ranks and their preservers

This paper concerns three notions of rank of matrices over semirings; real rank, semiring rank and column rank. These three rank functions are the same over subfields of reals but differ for matrices over subsemirings of nonnegative reals. We investigate the largest values of r for which the real rank and semiring rank, real rank and column rank of all m×n matrices over a given semiring are both r, respectively. We also characterize the linear operators which preserve the column rank of matrices over certain subsemirings of the nonnegative reals.