Essentially doubly stochastic matrices II. Canonical forms for row equivalence,equivalence, and a special case of congruence

In this paper we obtain canonical forms for row equivalence, equivalence, and a special case of congruence in the algebra of essentially doubly stochastic (e.d.s.) matrices of order n over a field F, char(F) [nmid] n. These forms are analogues of familiar forms in ordinary matrix algebra. The canonical form for equivalence is used in showing, in a subsequent paper, that every e.d.s. matrix of rank r and order n over F, char(F) = 0 or char(F) > n, is a product of elementary e.d.s. matrices, n – r of which are singular.     

Graph Mates

A weighted digraph graph D is said to be doubly stochastic if all the weights of the edges in D are in [0, 1] and sum of the weights of the edges incident to each vertex in D is one. Let Ω(G) be denoted as set of all doubly stochastic digraphs with n vertices. We defined a Graph Mates in Ω(G) and derived a necessary and sufficient condition for two doubly stochastic digraphs are to be a Graph Mates.     

A multidimensional Newton-Raphson method and its applications to the existence of asymptotic Fn-estimators and their stochastic expansions

Given a suitable function F_n we define a class of estimators called asymptotic F_n-estimators (i.e., estimators which approximate the solution of F_n(θ) = 0). It is proved that this class is nonvoid if appropriate regularity conditions are fulfilled and if one has at hand a suitable initial estimator. Furthermore, it is shown that F_n-estimators admit a stochastic expansion (which enables to give results on asymptotic expansions for the distribution of these estimators).     

Convex polyhedra of doubly stochastic matrices III. Affine and combinatorial properties of Ωn

Affine and combinatorial properties of the polytope $\text{Ω}{}_{\mathrm{n}}$ of all n × n nonnegative doubly stochastic matrices are investigated. One consequence of this investigation is that if $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$ of dimension d > 2, then $\text{F}$ has at most 3(d?1) facets. The special faces of $\text{Ω}{}_{\mathrm{n}}$ which were characterized in Part I of our study of $\text{Ω}{}_{\mathrm{n}}$ in terms of the corresponding (0, 1)- matrices are classified with respect to affine equivalence.     

p-Minors of a doubly stochastic matrix at which the permanent achieves a minimum

The author proves that if A is a matrix at which the permanent achieves a local minimum relative to the set of n x n doubly stochastic matrices, then for a_ij=0,

$\text{per A (i|j)?per A}$

.     

Vector Majorization Via Hessenberg Matrices

In this paper it is proved that, for real n-vectors x and y,x is majorized by y if and only if x = PHQy for some permutationmatrices P, Q, and for some doubly stochastic matrix H whichis a direct sum of doubly stochastic Hessenberg matrices. Thisresult reveals that any n-vector which is majorized by a vectory can be expressed as a convex combination of at most (n² –n + 2)/2 permutations of y.     

Spanning forests and the golden ratio

For a graph G, let f_ij be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path, the f_ij's can be expressed as the products of Fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers. The doubly stochastic graph matrix is the matrix F=(f_ij)_n×n/f, where f is the total number of spanning rooted forests of G and n is the number of vertices in G. F provides a proximity measure for graph vertices. By the matrix forest theorem, F^-1=I+L, where L is the Laplacian matrix of G. We show that for the paths and the so-called T-caterpillars, some diagonal entries of F (which provide a measure of the self-connectivity of vertices) converge to φ^-1 or to 1-φ^-1, where φ is the golden ratio, as the number of vertices goes to infinity. Thereby, in the asymptotic, the corresponding vertices can be metaphorically considered as “golden introverts” and “golden extroverts,” respectively. This metaphor is reinforced by a Markov chain interpretation of the doubly stochastic graph matrix, according to which F equals the overall transition matrix of a random walk with a random number of steps on G.     

On a conjecture of R.A. Brualdi

Brualdi's conjecture on the minimum permanent in the set of doubly stochastic n × n matrices with n − 1 zeros on a diagonal is shown to be false for n ⩾ 5. The minimum is determined in a subset of such matrices.     

Matrices satisfying the van der Waerden conjecture

It is shown here that any n×n doubly stochastic matrix whose numerical range lies in the sector from -π/2n to π/2n satisfies the van der Waerden conjecture.     

An extension of Birkhoff’s theorem with an application to determinants

Let M_n(F) denote the algebra of n×n matrices over the field F of complex, or real, numbers. Given a self-adjoint involution J∈M_n(C), that is, J=J^*,J²=I, let us consider Cⁿ endowed with the indefinite inner product [,] induced by J and defined by [x,y]?〈Jx,y〉,x,y∈Cⁿ. Assuming that (r,n-r), 0?r?n, is the inertia of J, without loss of generality we may assume J=diag(j₁,?,j_n)=I_r⊕-I_n-r. For T=(|t_ik|²)∈M_n(R), the matrices of the form T=(|t_ik|²j_ij_k), with all line sums equal to 1, are called J-doubly stochastic matrices. In the particular case r∈{0,n}, these matrices reduce to doubly stochastic matrices, that is, non-negative real matrices with all line sums equal to 1. A generalization of Birkhoff’s theorem on doubly stochastic matrices is obtained for J-doubly stochastic matrices and an application to determinants is presented.     

Convex polyhedra of doubly stochastic matrices: II. Graph of Ωn

Properties of the graph $\text{G(Ω}{}_{\mathrm{n}}\text{)}$ of the polytope $\text{Ω}{}_{\mathrm{n}}$ of all n × n nonnegative doubly stochastic matrices are studied. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$ which is not a k-dimensional rectangular parallelotope for k ≥ 2, then G($\text{F}$) is Hamilton connected. Prime factor decompositions of the graphs of faces of $\text{Ω}{}_{\mathrm{n}}$ relative to Cartesian product are investigated. In particular, if $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$, then the number of prime graphs in any prime factor decomposition of G($\text{F}$) equals the number of connected components of the neighborhood of any vertex of G($\text{F}$). Distance properties of the graphs of faces of $\text{Ω}{}_{\mathrm{n}}$ are obtained. Faces $\text{F}$ of $\text{Ω}{}_{\mathrm{n}}$ for which G($\text{F}$) is a clique of $\text{G(Ω}{}_{\mathrm{n}}\text{)}$ are investigated.     

Real permanental roots of doubly stochastic matrices

It is shown that for each n ? 7 there exists an n × n, irreducible, doubly stochastic matrix A such that the permanent of the characteristic matrix of A has n real zeros.     

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.     

Maximum and minimum diagonal sums of doubly stochastic matrices

Let Ω_n denote the convex polyhedron of all n×n doubly stochastic (d.s.) matrices. The purpose of this paper is to investigate some of the numerical properties of the maximum and the minimum diagonal sums of the matrices in Ω_n. A few conjectures that naturally arise will be mentioned.     

A threshold property for intersections in a finite set

Let F be any family of subsets of a finite set E and let n be an integer, n<|F|. Under what condition does the knowledge of cardinals of m-intersections in F, for all m≤n, univocally determine the cardinal of any intersection in F, and what is the minimal condition? We give a complete answer to that. For any n, this determination property is satisfied by n if and only if |E|<2ⁿ, without further condition on F.     

Subpermanents of doubly stochastic matrices †

It is shown that if all subpermaneats of order k of an n × n doubly stochastic matrix are equal for some k ≤ n ? 2, then all the entries of the matrix must be equal to 1/n.     

On entropy-preserving stochastic averages

When an n×n doubly stochastic matrix A acts on Rⁿ on the left as a linear transformation and P is an n-long probability vector, we refer to the new probability vector AP as the stochastic average of the pair (A,P). Let Γ_n denote the set of pairs (A,P) whose stochastic average preserves the entropy of P:H(AP)=H(P). Γ_n is a subset of B_n×Σ_n where B_n is the Birkhoff polytope and Σ_n is the probability simplex.We characterize Γ_n and determine its geometry, topology,and combinatorial structure. For example, we find that (A,P)∈Γ_n if and only if A^tAP=P. We show that for any n, Γ_n is a connected set, and is in fact piecewise-linearly contractible in B_n×Σ_n. We write Γ_n as a finite union of product subspaces, in two distinct ways. We derive the geometry of the fibers (A,·) and (·,P) of Γ_n. Γ₃ is worked out in detail. Our analysis exploits the convexity of xlogx and the structure of an efficiently computable bipartite graph that we associate to each ds-matrix. This graph also lets us represent such a matrix in a permutation-equivalent, block diagonal form where each block is doubly stochastic and fully indecomposable.     

On Some Zero Configurations Associated with the van der Waerden Conjecture

Let A be a permanent minimizing doubly stochastic matrix. This paper discusses the maximum number of zeros which can occur in any row or column of A. The results are applied to reaffirming the van der Waerden conjecture in the cases n?4.     

Doubly stochastic matrix equations

It is shown that for real,m x n matricesA andB the system of matrix equationsAX=B, BY=A is solvable forX andY doubly stochastic if and only ifA=BP for some permutation matrixP. This result is then used to derive other equations and to characterize the Green’s relations on the semigroup Ω _n of alln x n doubly stochastic matrices. The regular matrices in Ω _n are characterized in several ways by use of the Moore-Penrose generalized inverse. It is shown that a regular matrix in Ω _n is orthostochastic and that it is unitarily similar to a diagnonal matrix if and only if it belongs to a subgroup of Ω _n . The paper is concluded with extensions of some of these results to the convex setS _n of alln x n nonnegative matrices having row and column sums at most one. His research was supported by the N. S. F. Grant GP-15943.