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