Let $v_1 = (p_1,\dots,p_k)$ be a vector of interlacing polynomials, and non-negative coefficients, i.e. $p_1,\dots,p_k$ are real-rooted, and the roots of $p_i$, $p_{i+1}$ interlace.

Let $M$ be a $k \times k$ matrix with polynomial entries, and define $v_n = M^n v_0.$ For which $M$ is it true that $v_n$ consists of interlacing polynomials with non-negative coefficients, for all $n$?

There is a characterization of such matrices for $k=2$ in http://arxiv.org/abs/math/0612833 and some examples of such matrices for $k>2.$

What I need is some results/methods on how to see if a general $M$ preserves interlacing property. There are certain conditions that must be satisfied, such as the constant terms in the matrix gives only non-negative minors.

One such matrix, for example, is the matrix with 1 on and below the diagonal, and x above.

EDIT: Matrices as above preserving real-rooted polynomials with non-neg coefficients would also be helpful, (clearly if they preserve interlacing properties, then they preserve real-rootedness, but not always the other way around).