The scrambling index of symmetric primitive matrices

A nonnegative square matrix A is primitive if some power A^k>0 (that is, A^k is entrywise positive). The least such k is called the exponent of A. In [2], Akelbek and Kirkland defined the scrambling index of a primitive matrix A, which is the smallest positive integer k such that any two rows of A^k have at least one positive element in a coincident position. In this paper, we give a relation between the scrambling index and the exponent for symmetric primitive matrices, and determine the scrambling index set for the class of symmetric primitive matrices. We also characterize completely the symmetric primitive matrices in this class such that the scrambling index is equal to the maximum value.     

Maximal exponents of polyhedral cones (II)

Let K be a proper (i.e., closed, pointed, full convex) cone in Rⁿ. An n×n matrix A is said to be K-primitive if there exists a positive integer k such that ; the least such k is referred to as the exponent of A and is denoted by γ(A). For a polyhedral cone K, the maximum value of γ(A), taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ(K). It is proved that the maximum value of γ(K) as K runs through all n-dimensional minimal cones (i.e., cones having n+1 extreme rays) is n²-n+1 if n is odd, and is n²-n if n is even, the maximum value of the exponent being attained by a minimal cone with a balanced relation for its extreme vectors. The K-primitive matrices A such that γ(A) attain the maximum value are identified up to cone-equivalence modulo positive scalar multiplication.     

Monotonicity of permanents of certain doubly stochastic circulant matrices

Let p_k(A), k=2,…,n, denote the sum of the permanents of all k×k submatrices of the n×n matrix A. A conjecture of Ðokovi?, which is stronger than the famed van der Waerden permanent conjecture, asserts that the functions p_k((1?θ)J_n+;θA), k=2,…, n, are strictly increasing in the interval 0?θ?1 for every doubly stochastic matrix A. Here J_n is the n×n matrix all whose entries are equal $\text{1}\text{n}$. In the present paper it is proved that the conjecture holds true for the circulant matrices A=αI_n+ βP_n, α, β?0, α+;β=1, and $\text{A=}\text{(nJ}{}_{\mathrm{n}}\text{?I}{}_{\mathrm{n}}\text{?P}{}_{\mathrm{n}}\text{)}\text{(n?2)}$, where I_n and P_n are respectively the n×n identify matrix and the n×n permutation matrix with 1's in positions (1,2), (2,3),…, (n?1, n), (n, 1).     

A bound on the generalized competition index of a primitive matrix using Boolean rank

For a positive integer m where 1?m?n, the m-competition index (generalized competition index) of a primitive digraph is the smallest positive integer k such that for every pair of vertices x and y, there exist m distinct vertices v₁,v₂,…,v_m such that there are directed walks of length k from x to v_i and from y to v_i for 1?i?m. The m-competition index is a generalization of the scrambling index and the exponent of a primitive digraph. In this study, we determine an upper bound on the m-competition index of a primitive digraph using Boolean rank and give examples of primitive Boolean matrices that attain the bound.     

On the exponent of a primitive digraph

We characterize the equality case of the upper bound $\text{γ(D) ? n + s(n ? 2)}$ for the exponent of a primitive digraph in the case s ? 2, where n is the number of the vertices of the digraph D and s is the length of the shortest elementary circuit of D. We also answer a question about the equality case when s = 1. The exponent of an n × n primitive nonnegative matrix A is the same as the exponent of the associated digraph D(A) of A. So every theorem in this paper can be translated into a theorem about nonnegative matrices.     

Transposing a matrix by similarity operations

A matrix T is said to co-transpose a square matrix A if T^?1AT=A′ and T^?1A′T=A. For every n?3 there exists a real n×n matrix which cannot be co-transposed by any matrix. However, it is shown that the following classes of real matrices can be co-transposed by a symmetric matrix of order two: 2×2 matrices, normal matrices, and matrices whose square is symmetric.     

On a characterization of tridiagonal matrices by M. Fiedler

In this note two new proofs are given of the following characterization theorem of M. Fiedler: Let C_n, n?2, be the class of all symmetric, real matrices A of order n with the property that rank (A + D) ? n - 1 for any diagonal real matrix D. Then for any A ε C_n there exists a permutation matrix P such that PAP^T is tridiagonal and irreducible.     

Possible numbers of positive entries of imprimitive nonnegative matrices

Zhan, X., Extremal numbers of positive entries of imprimitive nonnegative matrix, Linear Algebra Appl. (in press) has determined the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices with a given imprimitivity index. Let σ( A ) denote the number of positive entries of a matrix A. Let M(n,?k) and m(n,?k) denote the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices of order n with a given imprimitivity index k, respectively. In this article, we prove that for any positive integer d with m(n,k)≤ d?≤?M(n,k), there exists an n?×?n irreducible nonnegative matrix A with imprimitivity index k such that?σ?(A)=d.     

Doubly stochastic matrices with some equal diagonal sums

Let A be doubly stochastic, and let τ₁,…,τ_m be m mutually disjoint zero diagonals in A, 1?m?n-1. E. T. H. Wang conjectured that if every diagonal in A disjoint from each τ_k (k=1,…,m) has a constant sum, then all entries in A off the m zero diagonals τ_k are equal to (n?m)^-1. Sinkhorn showed the conjecture to be correct. In this paper we generalize this result for arbitrary doubly stochastic zero patterns.     

On extremal matrices of second largest exponent by Boolean rank

Let b = b(A) be the Boolean rank of an n × n primitive Boolean matrix A and exp(A) be the exponent of A. Then exp(A) ? (b − 1)² + 2, and the matrices for which equality occurs have been determined in [D.A. Gregory, S.J. Kirkland, N.J. Pullman, A bound on the exponent of a primitive matrix using Boolean rank, Linear Algebra Appl. 217 (1995) 101-116]. In this paper, we show that for each 3 ? b ? n − 1, there are n × n primitive Boolean matrices A with b(A) = b such that exp(A) = (b − 1)² + 1, and we explicitly describe all such matrices.     

Products of reflections in the general linear group over a division ring

Let F be a division ring and A?GL_n(F). We determine the smallest integer k such that A admits a factorization A=R₁R₂?R_k?1B, where R₁,…,R_k?1 are reflections and B is such that rank(B?I_n)=1. We find that, apart from two very special exceptional cases, k=rank(A?I_n). In the exceptional cases k is one larger than this rank. The first exceptional case is the matrices A of the form I_m⊕αI_n?m where n?m?2, α≠?1, and α belongs to the center of F. The second exceptional case is the matrices A satisfying (A?I_n)²=0, rank(A?I_n)?2 in the case when char F≠2 only. This result is used to determine, in the case when F is commutative, the length of a matrix A?GL_n(F) with detA=±1 with respect to the set of all reflections in GL_n(F).     

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).     

On the separation power and the completion of partial latin squares

Let n and m be natural numbers, n ? m. The separation power of order n and degree m is the largest integer k = k(n, m) such that for every (0, 1)-matrix A of order n with constant linesums equal to m and any set of k 1's in A there exist (disjoint) permutation matrices P₁,…, P_m such that A = P₁ + … + P_m and each of the k 1's lies in a different P_i. Almost immediately we have 1 ? k(n, m) ? m ? 1, yet in all cases where the value of k(n, m) is actually known it equals m ? 1 (except under the somewhat trivial circumstances of k(n, m) = 1). This leads to a conjecture about the separation power, namely that k(n, m) = m ? 1 if $\text{m ? [}\text{n}\text{2}\text{] + 1}$. We obtain the bound $\text{k(n, m) ? m ? [}\text{n}\text{2}\text{] + 2}$, so that this conjecture holds for n ? 7. We then move on to latin squares, describing several equivalent formulations of the concept. After establishing a sufficient condition for the completion of a partial latin square in terms of the separation power, we can show that the Evans conjecture follows from this conjecture about the separation power. Finally the lower bound on k(n, m) allows us to show, after some calculations, that the Evans conjecture is true for orders n ? 11.     

Maximal exponents of polyhedral cones (I)

Let K be a proper (i.e., closed, pointed, full convex) cone in Rn. An n×n matrix A is said to be K-primitive if there exists a positive integer k such that ; the least such k is referred to as the exponent of A and is denoted by γ(A). For a polyhedral cone K, the maximum value of γ(A), taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ(K). It is proved that if K is an n-dimensional polyhedral cone with m extreme rays then for any K-primitive matrix A, γ(A)?(mA−1)(m−1)+1, where mA denotes the degree of the minimal polynomial of A, and the equality holds only if the digraph (E,P(A,K)) associated with A (as a cone-preserving map) is equal to the unique (up to isomorphism) usual digraph associated with an m×m primitive matrix whose exponent attains Wielandt's classical sharp bound. As a consequence, for any n-dimensional polyhedral cone K with m extreme rays, γ(K)?(n−1)(m−1)+1. Our work answers in the affirmative a conjecture posed by Steve Kirkland about an upper bound of γ(K) for a polyhedral cone K with a given number of extreme rays.     

Simultaneous block diagonalization of two real symmetric matrices

The simultaneous diagonalization of two real symmetric (r.s.) matrices has long been of interest. This subject is generalized here to the following problem (this question was raised by Dr. Olga Taussky-Todd, my thesis advisor at the California Institute of Technology): What is the first simultaneous block diagonal structure of a nonsingular pair of r.s. matrices ? For example, given a nonsingular pair of r.s. matrices S and T, which simultaneous block diagonalizations X′SX = diag(A₁, , A_k), X′TX = diag(B₁,, B_k) with dim A_i = dim B_i and X nonsingular are possible for 1 ? k ? n; and how well defined is a simultaneous block diagonalization for which k, the number of blocks, is maximal? Here a pair of r.s. matrices S and T is called nonsingular if S is nonsingular.If the number of blocks k is maximal, then one can speak of the finest simultaneous block diagonalization of S and T, since then the sizes of the blocks A_i are uniquely determined (up to permutations) by any set of generators of the pencil P(S, T) = {aS + bT|a, tb ε R} via the real Jordan normal form of S^?1T. The proof uses the canonical pair form theorem for nonsingular pairs of r.s. matrices. The maximal number k and the block sizes dim A_i are also determined by the factorization over C of ? (λ, μ) = det(λS + μT) for λ, μ ε R.     

Spaces of matrices of equal rank

Let M_m,n(F) denote the space of all mXn matrices over the algebraically closed field F. A subspace of M_m,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

$\text{UAV=}\text{A}{}_1\text{A}{}_2\text{A}{}_3\text{0}$

where A₁ is iX(k–i) for some i?k. Theorem: If $\text{K}$ is a space of rank k matrices, then either $\text{K}$ is essentially decomposable or dim $\text{K}$?k+1. An example shows that the above bound on non-essentially-decomposable spaces of rank k matrices is sharp whenever n?2k–1.     

Simultaneous triangularization of a pair of matrices whose commutator has rank two

Let A,B be n×n matrices with entries in an algebraically closed field F of characteristic zero, and let C=AB?BA. It is shown that if C has rank two and AⁱB^jC^k is nilpotent for 0?i, j?n?1, 1?k?2, then A, B are simultaneously triangularizable over F. An example is given to show that this result is in some sense best possible.     

On the span of Hadamard products of vectors

Let A₁, … , A_k be positive semidefinite matrices and B₁, … , B_k arbitrary complex matrices of order n. We show that

span{(A₁x)°(A₂x)°?°(A_kx)|x∈Cⁿ}=range(A₁°A₂°?°A_k)     

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.     

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.