Tridiagonal and Pentadiagonal Doubly Stochastic Matrices

We provide a decomposition that is sufficient in showing when a symmetric tridiagonal matrix A is completely positive and provide examples including how one can change the initial conditions or deal with block matrices, which expands the range of matrices to which our decomposition can be applied. Our decomposition leads us to a number of related results, allowing us to prove that for tridiagonal doubly stochastic matrices, positive semidefiniteness is equivalent to complete positivity (rather than merely being implied by complete positivity). We then consider symmetric pentadiagonal matrices, proving some analogous results, and providing two different decompositions sufficient for complete positivity, again illustrated by a number of examples

Tridiagonal and Pentadiagonal Doubly Stochastic Matrices

We provide a decomposition that is sufficient in showing when a symmetric tridiagonal matrix $A$ is completely positive and provide examples including how one can change the initial conditions or deal with block matrices, which expands the range of matrices to which our decomposition can be applied. Our decomposition leads us to a number of related results, allowing us to prove that for tridiagonal doubly stochastic matrices, positive semidefiniteness is equivalent to complete positivity (rather than merely being implied by complete positivity). We then consider symmetric pentadiagonal matrices, proving some analogous results, and providing two different decompositions sufficient for complete positivity, again illustrated by a number of examples.Comment: 15 page

Fastest random walk on a path

In this paper, we consider two convex optimisation problems in order to maximise the mixing rate of a Markov chain on an undirected path. In the first formulation, the holding probabilities of vertices are identical and the transition probabilities from a vertex to its neighbours are equal, whereas the second formulation is the more general reversible Markov chain with the same degree proportional stationary distribution. We derive analytical results on the solutions of the optimisation problems and compare the spectra of the associated transition probability matrices