On a Lie-theoretic approach to generalized doubly stochastic matrices and applications

In this article, we study generalized doubly stochastic matrices using the theory of Lie groups and Lie algebras. Applications to the inverse eigenvalue problem for symmetric doubly stochastic matrices are presented.     

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.     

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.     

Even and odd diagonals in doubly stochastic matrices

The existence of even or odd diagonals in doubly stochastic matrices depends on the number of positive elements in the matrix. The optimal general lower bound in order to guarantee the existence of such diagonals is determined, as well as their minimal number for given number of positive elements. The results are related to the characterization of even doubly stochastic matrices in connection with Birkhoff's algorithm.     

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.     

Minimum permanents on two faces of the polytope of doubly stochastic matrices

We consider the minimum permanents and minimising matrices on the faces of the polytope of doubly stochastic matrices whose nonzero entries coincide with those of, respectively,

     

Generalized Littlewood–Richardson rule and sum of coadjoint orbits of compact Lie groups

Let G be a compact connected semisimple Lie group. We extend to all irreducible finite-dimensional representations of G a result of Heckman which provides a relation between the generalized Littlewood–Richardson rule and the sum of G-coadjoint orbits. As an application of our result, we describe the eigenvalues of a sum of two real skew-symmetric matrices.     

Fibonacci numbers, alternating parity sequences and faces of the tridiagonal Birkhoff polytope

We determine the number of alternating parity sequences that are subsequences of an increasing m-tuple of integers. For this and other related counting problems we find formulas that are combinations of Fibonacci numbers. These results are applied to determine, among other things, the number of vertices of any face of the polytope of tridiagonal doubly stochastic matrices.     

Full ordering in the Shorrocks mobility sense of the semiring of monotone doubly stochastic matrices

In this paper, we investigate the ordering on a semiring of monotone doubly stochastic transition matrices in Shorrocks’ sense. We identify a class of an equilibrium index of mobility that induces the full ordering in a semiring, while this ordering is compatible with Dardanoni’s partial ordering on a set of monotone primitive irreducible doubly stochastic matrices.     

Minimal faithful representations of reductive Lie algebras

Let G and H be Lie groups with Lie algebras and . Let G be connected. We prove that a Lie algebra homomorphism is exact if and only if it is completely positive. The main resource is a corresponding theorem about representations on Hilbert spaces. This article summarizes the main results of [1]. Received: 6 December 2005     

Laplace’s method for iterated complex Brownian integrals

A kind of Laplace’s method is developped for iterated stochastic integrals where integrators are complex standard Brownian motions. Then it is used to extend properties of Bougerol and Jeulin’s path transform in the random case when simple representations of complex semisimple Lie algebras are not supposed to be minuscule.     

Weyl symbolic calculus on any Lie group

For any Lie groupG we extend the Weyl symbolic calculus to analytic symbol classes on the dual g^* of the Lie algebra g ofG.     

On the basic algebraic operations in solvable Lie groups

Summary For a simply connected solvable Lie group we specify the structure of the product, the inverse and the exponential map expressed in suitable coordinates (canonical coordinates of the second kind), and point out that in these coordinates the product and inverse are expressed entirely in terms of polynomials, exponential functions and trigonometric functions. We devise algorithms for computing the product, the inverse and the exponential map.     

Large time estimates for heat kernels in nilpotent Lie groups

The aim of this paper is to obtain some estimate for large time for the Heat kernel corresponding to a sub-Laplacian with drift term on a nilpotent Lie group. We also obtain a uniform Harnack inequality for a “bounded” family of sub-Laplacians with drift in the first commutator of the Lie algebra of the nilpotent group.     

Moving zeros among matrices

We investigate the zero-patterns that can be created by unitary similarity in a given matrix, and the zero-patterns that can be created by simultaneous unitary similarity in a given sequence of matrices. The latter framework allows a “simultaneous Hessenberg” formulation of Pati’s tridiagonal result for 4 × 4 matrices. This formulation appears to be a strengthening of Pati’s theorem. Our work depends at several points on the simplified proof of Pati’s result by Davidson and Djokovi?. The Hessenberg approach allows us to work with ordinary similarity and suggests an extension from the complex to arbitrary algebraically closed fields. This extension is achieved and related results for 5 × 5 and larger matrices are formulated and proved.     

Contributions to Real Exponential Lie Groups

 A Lie group is called exponential if its exponential map is surjective. It is called weakly exponential if the exponential image is dense, which is equivalent to the connectivity of each of the Cartan subgroups (compare [11]). In the present paper the authors study exponential Lie groups which are of mixed type, i.e., neither solvable nor semisimple. Necessary conditions and also, for special mixed Lie groups, sufficient conditions are given for exponentiality. Several counter examples are provided showing that none of the conditions which have surfaced during the course of our investigation can work as necessary and sufficient ones. All conditions considered deal with centralizers of ad-nilpotent elements of the Lie algebra. For example, it is shown that if G is exponential, there is a rather large characteristic subgroup B which contains the nilradical, all Levi factors, and all maximal compactly embedded subgroups, which is also exponential. Moreover, this subgroup is also Mal’cev splittable so that one can apply earlier results on Mal’cev splittable exponential Lie groups, which characterize exponentiality of these Lie groups (also by conditions concerning the centralizers of ad-nilpotent elements). (Received 1 June 1999; in final form 28 December 1999)     

Free local semigroup constructions

We introduce the notions of causal paths and causal homotopies, modifications of the traditional notions of paths and homotopies, as more suitable for certain basic constructions in (Lie) semigroup theory. The major result is the construction in this causal context of an analogue of the universal covering semigroup and the demonstration that local homomorphisms on the given semigroup extend to global homomorphisms on it. In certain important cases, it is shown that this semigroup actually agrees with the universal covering semigroup.The author gratefully acknowledges the partial support of theNational Science Foundation. DMS 910-4582.     

On pth roots of stochastic matrices

In Markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available. The problem arises of determining a stochastic pth root of a stochastic matrix (the given transition matrix). By exploiting the theory of functions of matrices, we develop results on the existence and characterization of matrix pth roots, and in particular on the existence of stochastic pth roots of stochastic matrices. Our contributions include characterization of when a real matrix has a real pth root, a classification of pth roots of a possibly singular matrix, a sufficient condition for a pth root of a stochastic matrix to have unit row sums, and the identification of two classes of stochastic matrices that have stochastic pth roots for all p. We also delineate a wide variety of possible configurations as regards existence, nature (primary or nonprimary), and number of stochastic roots, and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix.