Subtotally positive and Monge matrices

A real matrix is called k-subtotally positive if the determinants of all its submatrices of order at most k are positive. We show that for an m × n matrix, only mn inequalities determine such class for every k, 1 ? k ? min(m,n). Spectral properties of square k-subtotally positive matrices are studied. Finally, completion problems for 2-subtotally positive matrices and their additive counterpart, the anti-Monge matrices, are investigated. Since totally positive matrices are 2-subtotally positive as well, the presented necessary conditions for this completion problem are also necessary conditions for totally positive matrices.     

Preservers of pairs of bivectors with bounded distance

We extend Liu’s fundamental theorem of the geometry of alternate matrices to the second exterior power of an infinite dimensional vector space and also use her theorem to characterize surjective mappings T from the vector space V of all n×n alternate matrices over a field with at least three elements onto itself such that for any pair A, B in V, rank(A-B)?2k if and only if rank(T(A)-T(B))?2k, where k is a fixed positive integer such that n?2k+2 and k?2.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

Hermitian matrices with a bounded number of eigenvalues

Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate three by three Hermitian matrices is given, and the structure of the corresponding coordinate ring as a module over the special unitary group is determined. The method applies also for degenerate real symmetric three by three matrices. For arbitrary n   partial information on the minimal degree component of the vanishing ideal of the variety of n×n$n\times n$ Hermitian matrices with a bounded number of eigenvalues is obtained, and some known results on sum of squares presentations of subdiscriminants of real symmetric matrices are extended to the case of complex Hermitian matrices.     

Subspaces of matrices with special rank properties

Let K be a field and let M_m×n(K) denote the space of m×n matrices over K. We investigate properties of a subspace M of M_m×n(K) of dimension n(m-r+1) in which each non-zero element of M has rank at least r and enumerate the number of elements of a given rank in M when K is finite. We also provide an upper bound for the dimension of a constant rank r subspace of M_m×n(K) when K is finite and give non-trivial examples to show that our bound is optimal in some cases. We include a similar a bound for the maximum dimension of a constant rank subspace of skew-symmetric matrices over a finite field.     

Strong linear preservers of rank reverse permutability on triangular matrices

Let F be a field with ∣F∣ > 2 and T_n(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A₁, …, A_k ∈ T_n(F) is called rank reverse permutable if rank(A₁ A₂ ? A_k) = rank(A_k A_k−1 ? A₁). We characterize the linear maps on T_n(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.     

Characterizations of rectangular totally and strictly totally positive matrices

An n×m real matrix A is said to be totally positive (strictly totally positive) if every minor is nonnegative (positive). In this paper, we study characterizations of these classes of matrices by minors, by their full rank factorization and by their thin QR factorization.     

An upper bound for the minimum rank of a graph

For a graph G of order n, the minimum rank of G is defined to be the smallest possible rank over all real symmetric n×n matrices A whose (i,j)th entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs, including all bipartite graphs, and conjecture this bound is true over for all graphs, and prove a related bound for all zero-nonzero patterns of (not necessarily symmetric) matrices. Most of the results are valid for matrices over any infinite field, but need not be true for matrices over finite fields.     

Minimum rank of skew-symmetric matrices described by a graph

The minimum (symmetric) rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. The problem of determining minimum (symmetric) rank has been studied extensively. We define the minimum skew rank of a simple graph G to be the smallest possible rank among all skew-symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We apply techniques from the minimum (symmetric) rank problem and from skew-symmetric matrices to obtain results about the minimum skew rank problem.     

Determinantal varieties over truncated polynomial rings

We study components and dimensions of higher-order determinantal varieties obtained by considering generic m×n (m?n) matrices over rings of the form F[t]/(t^k), and for some fixed r, setting the coefficients of powers of t of all r×r minors to zero. These varieties can be interpreted as spaces of (k−1)th order jets over the classical determinantal varieties; a special case of these varieties first appeared in a problem in commuting matrices. We show that when r=m, the varieties are irreducible, but when r<m, these varieties are reducible. We show that when r=2<m (any k), there are exactly ⌊k/2⌋+1 components, which we determine explicitly, and for general r<m, we show there are at least ⌊k/2⌋+1 components. We also determine the components explicitly for k=2 and 3 for all values of r (for k=3 for all but finitely many pairs of (m,n)).     

Procrustes problems and associated approximation problems for matrices with k-involutory symmetries

We say that a matrix R∈C^n×n is k-involutary if its minimal polynomial is x^k-1 for some k?2, so R^k-1=R^-1 and the eigenvalues of R are 1,ζ,ζ²,…,ζ^k-1, where ζ=e^2πi/k. Let α,μ∈{0,1,…,k-1}. If R∈C^m×m, A∈C^m×n, S∈C^n×n and R and S are k-involutory, we say that A is (R,S,μ)-symmetric if RAS^-1=ζ^μA, and A is (R,S,α,μ)-symmetric if RAS^-α=ζ^μA.Let L be the class of m×n(R,S,μ)-symmetric matrices or the class of m×n(R,S,α,μ)-symmetric matrices. Given X∈C^n×t and B∈C^m×t, we characterize the matrices A in L that minimize ‖AX-B‖ (Frobenius norm), and, given an arbitrary W∈C^m×n, we find the unique matrix A∈L that minimizes both ‖AX-B‖ and ‖A-W‖. We also obtain necessary and sufficient conditions for existence of A∈L such that AX=B, and, assuming that the conditions are satisfied, characterize the set of all such A.     

Partial matrices whose completions have ranks bounded below

A partial matrix over a field F is a matrix whose entries are either elements of F or independent indeterminates. A completion of such a partial matrix is obtained by specifying values from F for the indeterminates. We determine the maximum possible number of indeterminates in a partial m×n matrix whose completions all have rank at least equal to a particular k, and we fully describe those examples in which this maximum is attained. Our main theoretical tool, which is developed in Section 2, is a duality relationship between affine spaces of matrices in which ranks are bounded below and affine spaces of matrices in which the (left or right) nullspaces of elements possess a certain covering property.     

Required nonzero patterns for nonsingular sign regular matrices

An n×n real matrix is called sign regular if, for each k(1?k?n), all its minors of order k have the same nonstrict sign. The zero entries which can appear in a nonsingular sign regular matrix depend on its signature because the signature can imply that certain entries are necessarily nonzero. The patterns for the required nonzero entries of nonsingular sign regular matrices are analyzed.     

Semigroup of matrices acting on the max-plus projective space

We investigate the action of semigroups of d×d matrices with entries in the max-plus semifield on the max-plus projective space. Recall that semigroups generated by one element with projectively bounded image are projectively finite and thus contain idempotent elements.In terms of orbits, our main result states that the image of a minimal orbit by an idempotent element of the semigroup with minimal rank has at most d! elements. Moreover, each idempotent element with minimal rank maps at least one orbit onto a singleton.This allows us to deduce the central limit theorem for stochastic recurrent sequences driven by independent random matrices that take countably many values, as soon as the semigroup generated by the values contains an element with projectively bounded image.     

On the dimension of some real spaces of bounded rank matrices

Given n∈N, let X be either the set of hermitian or real n×n matrices of rank at least n-1. If n is even, we give a sharp estimate on the maximal dimension of a real vector space V⊂X∪{0}. The results are obtained, via K-theory, by studying a bundle map induced by the adjunction of matrices.     

Construction of determinantal representation of trigonometric polynomials

For a pair of n×n Hermitian matrices H and K, a real ternary homogeneous polynomial defined by F(t,x,y)=det(tI_n+xH+yK) is hyperbolic with respect to (1,0,0). The Fiedler conjecture (or Lax conjecture) is recently affirmed, namely, for any real ternary hyperbolic polynomial F(t,x,y), there exist real symmetric matrices S₁ and S₂ such that F(t,x,y)=det(tI_n+xS₁+yS₂). In this paper, we give a constructive proof of the existence of symmetric matrices for the ternary forms associated with trigonometric polynomials.     

Real rank versus nonnegative rank

We consider the set of m×n nonnegative real matrices and define the nonnegative rank of a matrix A to be the minimum k such that A=BC where B is m×k and C is k×n. Given that the real rank of A is j for some j, we give bounds on the nonnegative rank of A and A².     

Maximum generic nullity of a graph

For a graph G of order n, the maximum nullity of G is defined to be the largest possible nullity over all real symmetric n×n matrices A whose (i,j)th entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Maximum nullity and the related parameter minimum rank of the same set of matrices have been studied extensively. A new parameter, maximum generic nullity, is introduced. Maximum generic nullity provides insight into the structure of the null-space of a matrix realizing maximum nullity of a graph. It is shown that maximum generic nullity is bounded above by edge connectivity and below by vertex connectivity. Results on random graphs are used to show that as n goes to infinity almost all graphs have equal maximum generic nullity, vertex connectivity, edge connectivity, and minimum degree.     

A system of real quaternion matrix equations with applications

Let H be the real quaternion algebra and H^n×m denote the set of all n×m matrices over H. Let P∈H^n×n and Q∈H^m×m be involutions, i.e., P²=I,Q²=I. A matrix A∈H^n×m is said to be (P,Q)-symmetric if A=PAQ. This paper studies the system of linear real quaternion matrix equations

     

Simultaneous solutions of Sylvester equations and idempotent matrices separating the joint spectrum

We investigate simultaneous solutions of the matrix Sylvester equations A_iX-XB_i=C_i,i=1,2,…,k, where {A₁,…,A_k} and {B₁,…,B_k} are k-tuples of commuting matrices of order m×m and p×p, respectively. We show that the matrix Sylvester equations have a unique solution X for every compatible k-tuple of m×p matrices {C₁,…,C_k} if and only if the joint spectra σ(A₁,…,A_k) and σ(B₁,…,B_k) are disjoint. We discuss the connection between the simultaneous solutions of Sylvester equations and related questions about idempotent matrices separating disjoint subsets of the joint spectrum, spectral mapping for the differences of commuting k-tuples, and a characterization of the joint spectrum via simultaneous solutions of systems of linear equations.