Matrices which commute with doubly stochastic matrices

The set doubly stochastic matrices which commute with the doubly stochastic matrices of any particular given rank is determined.     

Matrices which commute with doubly stochastic matrices

The set doubly stochastic matrices which commute with the doubly stochastic matrices of any particular given rank is determined.     

Essentially doubly stochastic matrices

In this paper we extend the general theory of essentially doubly stochastic (e.d.s.) matrices begun in earlier papers in this series. We complete the investigation in one direction by characterizing all of the algebra isomorphisms between the algebra of e.d.s. matrices of order n over a field F,E_n(F), and the total algebra of matrices of order n - 1over F,M_n-1(F) We then develop some of the theory when Fis a field with an involution. We show that for any e,f§Fof norm 1,e≠f every e.d.s. matrix in E_n(F) is a unique e.d.s. sum of an e.d.s. e-hermitian matrix and an e.d.s. f-hermitian matrix in E_n(F) Next, we completely determine the cases for which there exists an above-mentioned matrix algebra isomorphism preserving adjoints. Finally, we consider cogredience in E_n(F) and show that when such an adjoint-preserving isomorphism exists and char M_n(F) two e.d.s. e-hermitian matrices which are cogredient in M_n(F) are also cogredient in E_n(F). Using this result, we obtain simple canonical forms for cogredience of e.d.s. e-hermitian matrices in E_n(F) when Fsatisfies special conditions. This ncludes the e.d.s. skew-symmetric matrices, where the involution is trivial and E = -1.     

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.     

Permanents of doubly stochastic matrices

A conjecture on the permanents of doubly stochastic matrices is proposed. Some results supporting it are presented.     

Extreme symmetric doubly stochastic matrices

We characterize the extreme points of the polytope of symmetric doubly stochastic matrices of a given arbitrary order.     

Random doubly stochastic tridiagonal matrices

Let \begin{align*}{\mathcal T}\end{align*}n be the compact convex set of tridiagonal doubly stochastic matrices. These arise naturally in probability problems as birth and death chains with a uniform stationary distribution. We study ‘typical’ matrices T∈ \begin{align*}{\mathcal T}\end{align*}n chosen uniformly at random in the set \begin{align*}{\mathcal T}\end{align*}n. A simple algorithm is presented to allow direct sampling from the uniform distribution on \begin{align*}{\mathcal T}\end{align*}n. Using this algorithm, the elements above the diagonal in T are shown to form a Markov chain. For large n, the limiting Markov chain is reversible and explicitly diagonalizable with transformed Jacobi polynomials as eigenfunctions. These results are used to study the limiting behavior of such typical birth and death chains, including their eigenvalues and mixing times. The results on a uniform random tridiagonal doubly stochastic matrices are related to the distribution of alternating permutations chosen uniformly at random.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 403–437, 2013     

A k-measure of irreducibility and doubly stochastic matrices

A “k-measure of irreducibility” for a doubly stochastic matrix is developed with the aid of some combinatorial results. An application of this measure is established involving a lower bound for the second largest eigenvalue of a symmetric doubly stochastic matrix.     

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.     

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.     

A problem of Mirsky concerning nonsingular doubly stochastic matrices

The paper presents a new constructive proof of a theorem of Hardy, Littlewood, and Polya relating vector majorization and doubly stochastic matrices. Conditions on the vectors which guarantee that the corresponding matrices will be direct sums are given. These two results are applied to solve the problem, posed by Mirsky, of characterizing those majorization relations for which there is a corresponding doubly stochastic matrix which is nonsingular.     

A generalization of doubly stochastic matrices over a field

Let X and Y be m×n matrices over a field F such that Y^TX is nonsingular, and let Λ and Λ′ be sets of n-square matrices over F. Solutions A to the simultaneous equations AX = XK and $\text{Y}{}^{\mathrm{T}}\text{A =}\text{K}\text{?}\text{Y}{}^{\mathrm{T}}$ where K?Λ and $\text{K}\text{?}\text{? Λ′}$ are considered. It is shown that many properties of doubly stochastic matrices over a field have a natural generalization in terms of the set Δ(Λ,Λ′) of all such solutions.     

Convexity of the permanent for doubly stochastic matrices

Let Ω _n denote the set of all n×n doubly stochastic matrices and let J_n denote the n×n matrix all of whose entries are 1/n. Lih and Wang conjectuted that per[(1?i)J_n +iA≤(1?i)perJ_n 1i perA for all A∈Ω _n and all t∈[0,1/2], and proved their conjecture for n=3. In this paper we propose a similar conjecture asserting that for any A∈Ω _n \{J_n }, the permanent function is strietly convex on the straight line segment joining J_n and (J_n +A)/2, and prove it for the case n=3.     

On diagonal products of doubly stochastic matrices

The following result is proved: If A and B are distinct n × n doubly stochastic matrices, then there exists a permutation σ of {1, 2,…, n} such that ∏_ia_iσ(i) > ∏_ib_iσ(i).     

Convex sets of some doubly stochastic matrices

Let Ω denote the set of all n by n doubly stochastic matrices. Let t be a real number such that $\text{1}\text{t}\text{?}\text{1}\text{n}$ and let m be a real number such that $\text{1}\text{m}\text{? 1 ?}\text{1}\text{t}$. The set $\text{Ω}{}_{\mathrm{s}}\text{= {A ? Ω :}\text{1}\text{m}\text{? a}{}_{\mathrm{ij}}\text{?}\text{1}\text{t}\text{, 1 ? i, j ? n}}$ is the convex hull of the matrices in $\text{Ω}{}_{\mathrm{s}}$ having as many largest entries, namely, $\text{1}\text{t}$, as possible in each row and column while filling out the remaining entries with the value $\text{1}\text{m}$ and if necessary at most one entry in each row and column which has a value between $\text{1}\text{m}$ and $\text{1}\text{t}$.