The distance from a matrix polynomial to matrix polynomials with a prescribed multiple eigenvalue (Q947607)

For a given matrix polynomial \(P(\lambda )=\sum _{i=0}^mA_i\lambda ^i\), where the \(A_i\) are \(n\times n\) complex matrices, and for a given complex number \(\mu\) and integer \(\kappa \geq 2\), the authors determine the distance (suitably defined) from \(P(\lambda )\) to the set of matrix polynomials having \(\mu\) as an eigenvalue of geometric multiplicity at least \(\kappa\). They also obtain bounds for the distance from \(P(\lambda )\) to the set of matrix polynomials which have \(\mu\) as a multiple eigenvalue of any sort. This generalizes a result of \textit{A. N. Malyshev} [Numer. Math. 83, 443--454 (1999; Zbl 0972.15011)].