On Inverse Quadratic Eigenvalue Problems with Partially Prescribed Eigenstructure

Abstract

The inverse eigenvalue problem of constructing real and symmetric square matrices M, C, and K of size $n \times n$ for the quadratic pencil $Q(\lambda) = \lambda^2 M + \lambda C + K$ so that $Q(\lambda)$ has a prescribed subset of eigenvalues and eigenvectors is considered. This paper consists of two parts addressing two related but different problems. The first part deals with the inverse problem where M and K are required to be positive definite and semidefinite, respectively. It is shown via construction that the inverse problem is solvable for any k, given complex conjugately closed pairs of distinct eigenvalues and linearly independent eigenvectors, provided $k \leq n$. The construction also allows additional optimization conditions to be built into the solution so as to better refine the approximate pencil. The eigenstructure of the resulting $Q(\lambda)$ is completely analyzed. The second part deals with the inverse problem where M is a fixed positive definite matrix (and hence may be assumed to be the identity matrix $I_n$). It is shown via construction that the monic quadratic pencil $Q(\lambda)=\lambda^2 I_n + \lambda C + K$, with $n + 1$ arbitrarily assigned complex conjugately closed pairs of distinct eigenvalues and column eigenvectors which span the space $\mathbb{C}^n$, always exists. Sufficient conditions under which this quadratic inverse eigenvalue problem is uniquely solvable are specified.