Rate of stability of solutions of matrix polynomial and quadratic equations (Q1378068)

The rate of stability of solutions of matrix polynomial equations of the form \[ \sum^m_{i=0} A_iX^i=0, \tag{1} \] is studied, where \(A_i\in F^{n\times n}\), \(F=\mathbb{R}\) or \(F=\mathbb{C}\). A solution \(X\in F^{n\times n}\) is called \(\alpha\)-stable if there exist \(\varepsilon >0\), \(K>0\) such that every equation \(\sum^m_{i=0} B_iY^i=0\), \(B_i\in F^{n\times n}\) with \(\sum^m_{i=0} \| B_i- A_i\| <\varepsilon\) admits a solution \(Y\) with the property that \(\| Y-X \|<K \left(\sum^m_{i=0} \| B_i-A_i \|\right)^{1 \over\alpha}\) \((\alpha\) is said to be the rate of stability of \(X)\). The main result of \textit{A. C. M. Ran}, \textit{L. Rodman} and \textit{A. L. Rubin} [Linear Multilinear Algebra 36, No. 1, 27-39 (1993; Zbl 0798.15006)] is used, concerning \(\alpha\)-stable invariant subspaces. A full characterization of \(\alpha\)-stable solutions of (1) is obtained as a particular case of a more general framework of \(\alpha\)-stable factorizations of monic matrix polynomials. A special paragraph is dedicated to the case of weakly hyperbolic matrix polynomials \(L(\lambda)=\sum^m_{i=0}A_i\lambda^i\) i.e. which have the property that for every non-zero \(x\in F^n\) the scalar polynomial \(\langle L(\lambda)x,x\rangle\) has only real roots; \(\alpha\)-stable \((j\)-special) solutions with respect to Hermitian perturbations are defined and their characterization is given. Necessary and sufficient conditions are obtained for the existence of an \(\alpha\)-stable Hermitian solution for both continuous and discrete algebraic Riccati equations, as well as some characterizations of these solutions.