  The standard approach for finding eigenvalues and eigenvectors of matrix
polynomials starts by embedding the coefficients of the polynomial into a
matrix pencil, known as linearization. Building on the pioneering work of
Nakatsukasa and Tisseur, we present error bounds for the computed eigenvectors
of matrix polynomials. Our error bounds are applicable to any linearization
satisfying two properties. First, eigenvectors of the original matrix
polynomial can be recovered from those of the linearization without any
arithmetic computation. Second, the linearization presents one-sided
factorizations, which relate the residual for the linearization with the
residual for the polynomial. Linearizations satisfying these two properties
include the family of block Kronecker linearizations. The error bounds imply
that an eigenvector has been computed accurately when the residual norm is
small, provided that the computed associated eigenvalue is well-separated from
the rest of the spectrum of the linearization. The theory is illustrated by
numerical examples.

Numerical Analysis