Oscillation properties of generalized characteristic polynomials for totally positive and positive definite matrices

Oscillation Properties of Generalized Characteristic Polynomials for Totally Positive and Positive Definite Matrices

Fast algorithms for positive definite matrices structured by orthogonal polynomials

Positive definite matrices structured by orthogonal polynomial systems allow a Cholesky type decomposition of their inverse matrices in O(n2) steps. The algorithm presented in this paper uses the three-term recursion coefficients and the mixed moments of the involved polynomials.

Spectral Properties of Totally Positive Kernels and Matrices

We detail the history and present complete proofs of the spectral properties of totally positive kernels and matrices. §

Matrices of positive polynomials

Associated to a squarematrix all of whose entries are real Laurent polynomials in several variables with no negative coefficients is an ordered “dimension” module introduced by Tuncel, with additional structure, which acts as an invariant for topological Markov chains, and is also an invariant for actions of tori on AF C*-algebras. In describing this invariant, we are led naturally to eventuall...

Riemannian Sparse Coding for Positive Definite Matrices

Inspired by the great success of sparse coding for vector valued data, our goal is to represent symmetric positive definite (SPD) data matrices as sparse linear combinations of atoms from a dictionary, where each atom itself is an SPD matrix. Since SPD matrices follow a non-Euclidean (in fact a Riemannian) geometry, existing sparse coding techniques for Euclidean data cannot be directly extende...