On the distance from a matrix polynomial to matrix polynomials with some prescribed eigenvalues
Consider a matrix polynomial \(P(\lambda)=A_0+A_1\lambda+\dots+A_m\lambda^m\), where \(A_0,\dots,A_m\in\mathbb{C}^{n\times n}\) and \(A_m\) is invertible. Given distinct \(\mu_1,\dots,\mu_s\in\mathbb{C}\), consider the multiset \(\Sigma\) where \(\mu_i\) appears \(q_i\) (\(\geq 1\)) times, \(i=1,\dots,s\), and \(q_1+\dots+q_s=k\leq n\). The authors construct a matrix polynomial \(\Delta(\lambda)=\Delta_0+\Delta_1\lambda+\dots+\Delta_m\lambda^m\), \(\Delta_0,\dots,\Delta_m\in\mathbb{C}^{n\times n}\), such that the spectrum of \(Q(\lambda)=P(\lambda)+\Delta(\lambda)\) contains \(\Sigma\). The special case \(s=k\) has previously been settled in [the first author et al., Linear Multilinear Algebra 65, No. 4, 658--676 (2017; Zbl 1360.15013)] and the case \(s=1\) in [\textit{P. J. Psarrakos}, Linear Algebra Appl. 436, No. 11, 4107--4119 (2012; Zbl 1252.15025)]. Given \(\varepsilon\geq 0\) and \(w=\{w_0,\dots,w_m\}\) with \(w_0>0\), \(w_1,\dots,w_m\geq 0\), let \[ \mathcal{B}(P,\varepsilon,w)= \{Q(\lambda)\mid\|\Delta_j\|_2\leq\varepsilon w_j,\,j=0,\dots,m\}, \] where \(\|\cdot\|_2\) stands for the spectral norm. The spectral norm distance from \(P(\lambda)\) to the set of matrix polynomials whose spectra include \(\Sigma\) is defined by \[ D_w(P,\Sigma)=\min\{\varepsilon\geq 0\mid\exists \, Q(\lambda)\in\mathcal{B}(P,\varepsilon,w):\Sigma\subseteq\sigma(Q)\}, \] where \(\sigma(Q)\) denotes the spectrum of \(Q\). The authors present bounds for \(D_w(P,\Sigma)\).
scientific article; zbMATH DE number 6883433
scientific article