Tridiagonalization of completely nonnegative matrices

Abstract:

Let $M = [{m_{ij}}]_{i,j = 1}^n$ be completely nonnegative (CNN), i.e., every minor of $M$ is nonnegative. Two methods for reducing the eigenvalue problem for $M$ to that of a CNN, tridiagonal matrix, $T = [{t_{ij}}]$ (${t_{ij}} = 0$ when $|i - j| > 1)$), are presented in this paper. In the particular case that $M$ is nonsingular it is shown for one of the methods that there exists a CNN nonsingular $S$ such that $SM = TS$.