  In this paper, we plan to show an eigenvalue algorithm for block Hessenberg
matrices by using the idea of non-commutative integrable systems and
matrix-valued orthogonal polynomials. We introduce adjacent families of
matrix-valued $\theta$-deformed bi-orthogonal polynomials, and derive
corresponding discrete non-commutative hungry Toda lattice from discrete
spectral transformations for polynomials. It is shown that this discrete system
can be used as a pre-precessing algorithm for block Hessenberg matrices.
Besides, some convergence analysis and numerical examples of this algorithm are
presented.
