On Some Inverse Eigenvalue Problems of Quadratic Palindromic Systems

Abstract

This paper concerns some inverse eigenvalue problems of the quadratic $\star$-(anti)-palindromic system $Q(\lambda )={\lambda }^2{A}_1^{\star }+\lambda A_0+\epsilon A_1$, where $\epsilon =\pm 1$, $A_1,A_0\in {\mathbb{C}}^{n\times n}$, $A_0^{\star }=\epsilon A_0$, $A_1$ is nonsingular, and the symbol $\star$ is used as an abbreviation for transpose for real matrices and either transpose or conjugate transpose for complex matrices. By using the spectral decomposition of the quadratic $\star$-(anti)-palindromic system, the inverse eigenvalue problems with entire/partial eigenpairs given, and the model updating problems with no-spillover are considered. Some conditions on the solvabilities of these problems are given, and algorithms are proposed to find these solutions. These algorithms are illustrated by some numerical examples.