Sign patterns,nonsingularity, and the solvability of Ax = b

We investigate conditions on the sign-pattern class of the (n–1)st compound of a real n-by-n matrix A such that the solvability of Ax = b⁽ⁱ⁾ for i = 1,…,k, k<n, with specific b⁽ⁱ⁾, insures the nonsingularity of A. The number and choice of right-hand sides b⁽ⁱ⁾ sufficient for the task depends only on the sign-pattern class of the (n–1)st compound of A. The result for k = 1 generalizes a known fact about totally nonnegative matrices and an observation about M-matrices, thus providing another unifying result for these two classes of matrices.     

The primitive Boolean matrices with the second largest scrambling index by Boolean rank

The scrambling index of an n × n primitive Boolean matrix A is the smallest positive integer k such that A ^k (A ^T) ^k = J, where A ^T denotes the transpose of A and J denotes the n×n all ones matrix. For an m×n Boolean matrix M, its Boolean rank b(M) is the smallest positive integer b such that M = AB for some m × b Boolean matrix A and b×n Boolean matrix B. In 2009, M. Akelbek, S. Fital, and J. Shen gave an upper bound on the scrambling index of an n×n primitive matrix M in terms of its Boolean rank b(M), and they also characterized all primitive matrices that achieve the upper bound. In this paper, we characterize primitive Boolean matrices that achieve the second largest scrambling index in terms of their Boolean rank.     

On the Perron-Frobenius eigenvector for nonnegative integral matrices whose largest eigenvalue is integral

We first derive the bound |det(λI − A)|⩽λ^k − λ^k₀ (λ₀⩽λ), where A is a k × k nonnegative real matrix and λ₀ is the spectral radius of A. If A is irreducible and integral, and its largest nonnegative eigenvalue is an integer n, then we use this inequality to derive the upper bound n^k−1 on the components of the smallest integer eigenvector corresponding to n. Finer information on the components is also derived.     

Possible isolation number of a matrix over nonnegative integers

Let ?₊ be the semiring of all nonnegative integers and A an m × n matrix over ?₊. The rank of A is the smallest k such that A can be factored as an m × k matrix times a k×n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of A. For A with isolation number k, we investigate the possible values of the rank of A and the Boolean rank of the support of A. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of m×n matrices whose isolation number is m. That is, those matrices are permutationally equivalent to a matrix A whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.     

A relationship between stochastic matrices and eigenvalues of products

Let {B₁,…,B_n} be a set of n rank one n×n row stochastic matrices. The next two statements are equivalent: (1) If A is an n×n nonnegative matrix, then 1 is an eigenvalue ofB_kA for each k=1,…,n if and only if A is row stochastic. (2) The n×n row stochastic matrix S whose kth row is a row of B_k for k=1,…,n is nonsingular. For any set {B₁, B₂,…, B_p} of fewer than n row stochastic matrices of order n×n and of any rank, there exists a nonnegative n×n matrix A which is not row stochastic such that 1 is eigenvalue of every B_kA, k=1,…,p.     

A matrix of permanents

If A is a matrix of order n×(n?2), n?3, denote by ā the n×n matrix whose (i,j)th entry is zero if i=j, and if i≠j, is the permanent of the submatrix of A obtained by deleting its ith and jth rows. It is shown that if A is a nonnegative n×(n?2) matrix, then ā is nonsingular if and only if A has no zero submatrix of n?1 lines. This is used to give precise consequences of the occurrence of equality in Alexandroff's inequality.     

Permanents of d-dimensional matrices

Let A = (A_i1i2…_id) be an n₁ × n₂ × · × n_d matrix over a commutative ring. The permanent of A is defined by per (A) = ∑πⁿ_1i = ₁A_iσ2(_i)σ₃(_i)…σ_d(_i), where the summation ranges over all one-to-one functions σ_k from {1,2,…, n₁} to {1,2,…, n_k}, k = 2,3,…, d. In this paper it is shown that a number of properties of permanents of 2-dimensional matrices extend to higher-dimensional matrices. In particular, permanents of nonnegative d-dimensional matrices with constant hyperplane sums are investigated. The paper concludes by introducing s-permanents, which generalize the definition above that the permanent becomes the 1-permanent, and showing that an s-permanent can always be converted into a 1-permanent.     

Sign-nonsingular matrices and even cycles in directed graphs

A sign-nonsingular matrix or L-matrix A is a real m× n matrix such that the columns of any real m×n matrix with the same sign pattern as A are linearly independent. The problem of recognizing square L-matrices is equivalent to that of finding an even cycle in a directed graph. In this paper we use graph theoretic methods to investigate L-matrices. In particular, we determine the maximum number of nonzero elements in square L-matrices, and we characterize completely the semicomplete L-matrices [i.e. the square L-matrices (a_ij) such that at least one of a_ij and a_ij is nonzero for any i,j] and those square L-matrices which are combinatorially symmetric, i.e., the main diagonal has only nonzero entries and a_ij=0 iff a_ji=0. We also show that for any n×n L-matrix there is an i such that the total number of nonzero entries in the ith row and ith column is less than n unless the matrix has a completely specified structure. Finally, we discuss the algorithmic aspects.     

A combinatorial problem involving graphs and matrices

In this paper we discuss a combinatorial problem involving graphs and matrices. Our problem is a matrix analogue of the classical problem of finding a system of distinct representatives (transversal) of a family of sets and relates closely to an extremal problem involving 1-factors and a long standing conjecture in the dimension theory of partially ordered sets. For an integer n ?1, let n denote the n element set {1,2,3,…, n}. Then let A be a k×t matrix. We say that A satisfies property P(n, k) when the following condition is satisfied: For every k-taple (x₁,x₂,…,x_k?n^k there exist k distinct integers j₁,j₂,…,j_k so that x_i= a_ii for i= 1,2,…,k. The minimum value of t for which there exists a k × t matrix A satisfying property P(n,k) is denoted by f(n,k). For each k?1 and n sufficiently large, we give an explicit formula for f(n, k): for each n?1 and k sufficiently large, we use probabilistic methods to provide inequalities for f(n,k).     

Simple product colorings

If A is a set colored with m colors, and B is colored with n colors, the coloring of A × B obtained by coloring (a, b) with the pair (color of a, color of b) will be called an m × n simple product coloring (SPC) of A × B. SPC's of Cartesian products of three or more sets are defined analogously. It is shown that there are 2 × 2, and 2 × 2 × 2 SPC's of $\text{Q}$² and $\text{Q}$³ which forbid the distance one; that there is no 2^k SPC of $\text{Q}$^k forbidding the distance one, for k > 3; and that there is no 2 × 2 SPC of $\text{Q}$ × $\text{Q}$(√15), and thus none of $\text{R}$², forbidding the distance 1.     

A subdeterminant inequality for normal matrices

Let A be an n × n normal matrix over $\text{C}$, and Q_m, _n be the set of strictly increasing integer sequences of length m chosen from 1,…,n. For α, β ? Q_m, _n denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k ? {0, 1,…, m} we write |α∩β| = k if there exists a rearrangement of 1,…, m, say i₁,…, i_k, i_k+1,…, i_m, such that α(i_j) = β(i_j), i = 1,…, k, and {α(i_k+1),…, α(i_m) } ∩ {β(i_k+1),…, β(i_m) } = ?. A new bound for |detA[α|β ]| is obtained in terms of the eigenvalues of A when 2m = n and |α∩β| = 0.Let $\text{U}$_n be the group of n × n unitary matrices. Define the nonnegative number

where | α ∩ β| = k. It is proved that

Let A be semidefinite hermitian. We conjecture that ρ₀(A) ? ρ₁(A) ? ··· ? ρ_m(A). These inequalities have been tested by machine calculations.     

Overdetermined linear systems satisfying nonnegativity constraints

Let A be a nonnegative m × n matrix, and let b be a nonnegative vector of dimension m. Also, let S be a subspace of $\text{R}$ⁿ such that if P_S is the orthogonal projector onto S, then P_S ? 0. A necessary condition is given for the matrix A to satisfy the following property: For all b ? 0, if min[boxV]b ? Ax[boxV] is attained at x = x₀, then x₀ ? 0 and x₀ ? S. It is also shown that if a nonnegative matrix A has a nonnegative generalized inverse, then any submatrix of A also possesses a nonnegative generalized inverse.     

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.     

Utilizing Shelve Slots: Sufficiency Conditions for Some Easy Instances of Hard Problems

In this paper we present a minimal set of conditions sufficient to assure the existence of a solution to a system of nonnegative linear diophantine equations. More specifically, suppose we are given a finite item set U = {u₁, u₂, . . . , u_k} together with a "size" v_i ≡ v(u_i) ∈ Z⁺, such that v_i ≠ v_j for i ≠ j, a "frequency" a_i ≡ a(u_i) ∈ Z⁺, and a positive integer (shelf length) L ∈ Z⁺ with the following conditions: (i) L = ∏ⁿ_j=1p_j(p_j ∈ Z⁺ ∀j, p_j ≠ p_l for j ≠ l) and v_i = ∏ _j∈Aip_j, A_i ⊆ {l, 2, . . . , n} for i = 1, . . . , n; (ii) (A_i\{⋂^k_j=1A_j}) ∩ (A_l\{⋂^k_j=1A_j}) = ⊘∀i ≠ l. Note that v_i|L (divides L) for each i. If for a given m ∈ Z⁺, ∑ⁿ_i=1a_iv_i = mL (i.e., the total size of all the items equals the total length of the shelf space), we prove that conditions (i) and (ii) are sufficient conditions for the existence of a set of integers {b₁₁, b₁₂, . . . , b_1m, b₂₁, . . . , b_n1, . . . , b_nm}⊆ N such that ∑^m_j=1b_ij = a_i, i = 1, . . . , k, and ∑^k_i=1b_ijv_i = L, j =1, . . . , m (i.e., m shelves of length L can be fully utilized). We indicate a number of special cases of well known NP-complete problems which are subsequently decided in polynomial time.     

Generating all linear transformations

For an n×n Boolean matrix R, let A_R={n×n matrices A over a field F such that if r_ij=0 then a_ij=0}. We show that a collection A_R〈1〉,…,A_R〈k〉 generates all n×n matrices over F if and only if the matrix J all of whose entries are 1 can be expressed as a Boolean product of Hall matrices from the set {R〈1〉,…,R〈k〉}. We show that J can be expressed as a product of Hall matrices R〈i〉 if and only if ΣR〈i〉?R〈i〉 is primitive.     

An extremal property of the permanent and the determinant

Given an n × n matrix A, define the n! × n! matrix Ã, with rows and columns indexed by the permutation group S_n, with (σ, τ) entry Πⁿ_i=1a_{τ(i), σ(i)}. It is shown that if A is positive semidefinite, then det A is the smallest eigenvalue of Ã; it is conjectured that per A is the largest eigenvalue of Ã, and the conjecture proved for n⩽3. Several known and some unknown inequalities are derived as consequences.     

Monotonicity of permanents of direct sums of doubly stochastic matrices

Let Ωn be the set of all n × n doubly stochastic matrices, let J_n be the n × n matrix all of whose entries are 1/n and let σ _k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω _n and A ≠ J_J then g_A,k (θ) ? σ _k ((1 θ)J_n 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A ₁ ⊕ ⊕A_t (t ≥ 2) is an n × n matrix where A_i for i = 1,2, …,t, and if for each i g_Ai,ki (θ) is non-decreasing on [0.1] for k_t = 2,3,…,n_i , then g_A,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n.     

A generalization of the fast LUP matrix decomposition algorithm and applications

We show that any m × n matrix A, over any field, can be written as a product, LSP, of three matrices, where L is a lower triangular matrix with l's on the main diagonal, S is an m × n matrix which reduces to an upper triangular matrix with nonzero diagonal elements when the zero rows are deleted, and P is an n × n permutation matrix. Moreover, L, S, and P can be found in O(m^α?1n) time, where the complexity of matrix multiplication is O(m^α). We use the LSP decomposition to construct fast algorithms for some important matrix problems. In particular, we develop O(m^α?1n) algorithms for the following problems, where A is any m × n matrix: (1) Determine if the system of equations $\text{A}\text{x}\text{=}\text{b}$ (where $\text{b}$ is a column vector) has a solution, and if so, find one such solution. (2) Find a generalized inverse, $\text{A}{}^1$, of A (i.e., $\text{AA}{}^1\text{A = A}$). (3) Find simultaneously a maximal independent set of rows and a maximal independent set of columns of A.     

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].     

Asymptotics of products of nonnegative random matrices

Asymptotic properties of products of random matrices ξ _k = X _k …X ₁ as k → ∞ are analyzed. All product terms X _i are independent and identically distributed on a finite set of nonnegative matrices A = {A ₁, …, A _m }. We prove that if A is irreducible, then all nonzero entries of the matrix ξ _k almost surely have the same asymptotic growth exponent as k→∞, which is equal to the largest Lyapunov exponent λ(A). This generalizes previously known results on products of nonnegative random matrices. In particular, this removes all additional “nonsparsity” assumptions on matrices imposed in the literature.We also extend this result to reducible families. As a corollary, we prove that Cohen’s conjecture (on the asymptotics of the spectral radius of products of random matrices) is true in case of nonnegative matrices.