On Some Inverse Eigenvalue Problems of Quadratic Palindromic Systems

This paper concerns some inverse eigenvalue problems of the quadratic $\star$-(anti)-palindromic system $Q(λ)=λ^2 A_1^{\star}+λA_0 + εA_1$, where $ε=\pm 1$, $A_1, A_0 \in \mathbb{C}^{n\times n}$, $A_0^{\star}=εA_0$, $A_1$ 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.