Triangularizing Quadratic Matrix Polynomials

Abstract

We show that any regular quadratic matrix polynomial can be reduced to an upper triangular quadratic matrix polynomial over the complex numbers preserving the finite and infinite elementary divisors. We characterize the real quadratic matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable are quasi-triangularizable with diagonal blocks of sizes $1\times 1$ and $2 \times 2$. We also derive complex and real Schur-like theorems for linearizations of quadratic matrix polynomials with nonsingular leading coefficients. In particular, we show that for any monic linearization $\lambda I+A$ of an $n\times n$ quadratic matrix polynomial there exists a nonsingular matrix defined in terms of $n$ orthonormal vectors that transforms $A$ to a companion linearization of a (quasi-)triangular quadratic matrix polynomial. This provides the foundation for designing numerical algorithms for the reduction of quadratic matrix polynomials to upper (quasi-)triangular form.