Verified Computation of Eigenpairs in the Generalized Eigenvalue Problem for Nonsquare Matrix Pencils

Abstract:

      Consider an optimization problem arising from the generalized eigenvalue problem $Ax = \lambda Bx$, where $A,B \in \mathbb{C}^{m \times n}$ and $m > n$. Ito et al. showed that the optimization problem can be solved by utilizing right singular vectors of $C := [B,A]$. In this paper, we focus on computing intervals containing the solution. When some singular values of $C$ are multiple or nearly multiple, we can enclose bases of corresponding invariant subspaces of $C^HC$, where $C^H$ denotes the conjugate transpose of $C$, but cannot enclose the corresponding right singular vectors. The purpose of this paper is to prove that the solution can be obtained even when we utilize the bases instead of the right singular vectors. Based on the proved result, we propose an algorithm for computing the intervals. Numerical results show property of the algorithm.