On Householder sets for matrix polynomials

We present a generalization of Householder sets for matrix polynomials. After defining these sets, we analyze their topological and algebraic properties, which include containing all of the eigenvalues of a given matrix polynomial. Then, we use instances of these sets to derive the Geršgorin set, weighted Geršgorin set, and weighted pseudospectra of a matrix polynomial. Finally, we show that Householder sets are intimately connected to the Bauer-Fike theorem by using these sets to derive Bauer-Fike-type bounds for matrix polynomials.