A study of the van der Waerden conjecture and its generalizations

It Is shown here that a permanent of n× ndoubly stochastic matrix is not less than 1/ n!. The proof of this inequality follows from the study of a generalized van der Waerden problem on the set of doubly stochastic matrices.     

Minimum permanents of doubly stochastic matrices with zero main diagonal

The permanent function on the set of n×n doubly stochastic matrices with zero main diagonal attains a strict local minimum at the matrix whose off diagonal entries are all equal to 1/(n-1).     

On the permanent of doubly stochastic matrices with zero diagonal

The permanent function on the set of n×n doubly stochastic matrices with zero main diagonaln≤4, attains its minimum uniquely at the matrix whose off-diagonal entries are all equal to l/(n-1).     

A characterization of linear transformations which leave the doubly stochastic matrices

A characterization of linear transformations which leave the n×n doubly stochastic matrices invariant is given as a linear combination of functions of the type T(X)=AXB with certain restrictions posed on the n×n matrices A and B.     

Minimum permanents of doubly stochastic matrices with prescribed zero entries

A study of properties of matrices with minimum permanent in a face of the polyhedron of doubly stochastic n × n matrices. The minima are determined for certain faces.     

Row Stochastic Matrices Similar to Doubly Stochastic Matrices

The problem of determining which row stochastic n-by-n matrices are similar to doubly stochastic matrices is considered. That not all are is indicated by example, and an abstract characterization as well as various explicit sufficient conditions are given. For example, if a row stochastic matrix has no entry smaller than (n+1)^-1 it is similar to a doubly stochastic matrix.

Relaxing the nonnegativity requirement, the real matrices which are similar to real matrices with row and column sums one are then characterized, and it is observed that all row stochastic matrices have this property. Some remarks are then made on the nonnegative eigenvalue problem with respect to i) a necessary trace inequality and ii) removing zeroes from the spectrum.     

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.     

Nonchordal positive semidefinite stochastic matrices

Let K_nbe the convex set of n×npositive semidefinite doubly stochastic matrices. If Aε k_n, the graph of A,G(A), is the graph on n vertices with (i,j) an edge if a_ij ≠ 0i≠ j. We are concerned with the extreme points in K_n. In many cases, the rank of Aand G(A) are enough to determine whether A is extreme in K_n. This is true, in particular, if G(A)is a special kind of nonchordal graph, i.e., if no two cycles in G(A)have a common edge.     

Minimum permanents of doubly stochastic matrices with at least one zero entry

It is shown that the minimum value of the permanent on the n× ndoubly stochastic matrices which contain at least one zero entry is achieved at those matrices nearest to J_nin Euclidean norm, where J_nis the n× nmatrix each of whose entries is n^-1. In case n ≠ 3 the minimum permanent is achieved only at those matrices nearest J_n; for n= 3 it is achieved at other matrices containing one or more zero entries as well.     

Products of elementary doubly stochastic matrices

While studying a theorem of Westwerk on higher numerical ranges, we became interested in how the theory of elementary doubly stochastic (e.d.s.) matrices is related to a result of Goldberg and Straus. We show that there exist classes of doubly stochastic (d.s.) matrices of order n≧3 and orthostochastic (o s) matrices of order n≧4 such that the matrices in these classes cannot be represented as a product of e.d.s. matrices. In fact the matrices in these classes do not admit a representation as an infinite limit of a product of e.d.s. matrices.     

On two permanental conjectures

In this paper, we present some properties of the permanent of doubly stochastic matrices, and we prove that a conjecture of T. H. Foregger and a conjecture of B. Gyires, both on permanents, hold for n = 3.     

Permanents of doubly stochastic matrices with fixed zero pattern

The doubly stochastic matrices with a given zero pattern which are closest in Euclidean norm to J_nn, the matrix with each entry equal to 1/n, are identified. If the permanent is restricted to matrices having a given zero pattern confined to one row or to one column, the permanent achieves a local minimum at those matrices with that zero pattern which are closest to J_nn. This need no longer be true if the zeros lie in more than one row or column.     

A Note on degree antiregular graphs

This note proves a conjecture of Merris that the minimal value of entries of the doubly stochastic matrix of the degree antiregular graph E_n of order n ≥ 3 is equal to (l/2(n + l)).     

The doubly stochastic matrix equations x:aty= A and X Aty=t

For a doubly stochastic matrix A, each of the equations x:a^ty= A and X A^ty=^t is shown to have doubly stochastic solutions X and Y if and only if A lies in a subgroup of the semigroup of all doubly stochastic matrices of a given order. All elements of this semigroup which are left regular, right regular, or intra-regular are identified.     

Doubly stochastic matrices whose powers eventually stop

In this note we characterize doubly stochastic matrices A whose powers A,A²,A³,… eventually stop, i.e., A^p=A^p+1= for some positive integer p. The characterization enables us to determine the set of all such matrices.     

Between d-stability and sign-stability: the 3 × 3 case

[2] introduced a decreasing sequence of sets of real n × n matrices, which begins with the D-stable matrices and stops at the sign-stable matrices. It is not clear how many of the n sets in the sequence are distinct. This article documents the disappointment that in the first case where the sequence could contain a set which is neither the D-stable matrices nor the sign-stable matrices(viz., the case n = 3) it doesn't.     

Some rationally looking faces of Ωn having irrational minimum permanents

Let Ω_n denote the convex polytope consisting of all n × n doubly stochasiic matrices. We determine the minimum permanents which may or may not be rational and the permanent-minimizing matrices over some rationally looking faces of Ω_n We also discuss the barycentricity of the (0, l)-matrices with which we consider the permanent-minimization problem.     

Commuting pairs and triples of matrices and related varieties

In this note, we show that the set of all commuting d-tuples of commuting n×n matrices that are contained in an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber's theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs. We show that the variety of commuting triples of 4×4 matrices is irreducible. We also study the variety of n-dimensional commutative subalgebras of M_n(F), and show that it is irreducible of dimension n²−n for n4, but reducible, of dimension greater than n²−n for n7.     

The Best Lower Bound for the Permanent of a Correlation Matrix of Rank Two

If 1≤k≤n, then Cor(n,k) denotes the set of all n×n real correlation matrices of rank not exceeding k. Grone and Pierce have shown that if A∈Cor (n, n-1), then per(A)≥n/(n-1). We show that if A∈Cor(n,2), then , and that this inequality is the best possible.     

Some problems and results concerning the spectrums of stochastic matrices

Let Abe an n× nnonnegative matrix. Under suitable conditions there exist diagonal matrices Dand Ewith positive main diagonals such that DAEis doubly stochastic. In this paper we discuss some problems concerning the spectrums of Aand DAEwhen Ais row stochastic.