On factorization of matrix polynomials (Q1970497)

A matrix polynomial \(P(\lambda)\) is a polynomial, where the coefficients are \(n\times n\) matrices with complex entries. The numerical range of \(P(\lambda)\) is the set of all \(\lambda\in{\mathbb{C}}\) such that \(x^\star P(\lambda)x=0\) for some nonzero \(x\in{\mathbb{C}}^n.\) The authors show, for example, that if the numerical range of a matrix polynomial has exactly \(\rho\) bounded and connected components, then under certain additional conditions \(P(\lambda)\) has a spectral factorization into \(\rho\) suitable matrix polynomials. The authors provide two examples, they discuss the maximal number of connected components of a numerical range, and they treat quadratic matrix polynomials in more detail.