Generalized Eigenvalues of Nonsquare Pencils with Structure
Pablo Lecumberri, Marisol Gómez, A. Carlosena · SIAM Journal on Matrix Analysis and Applications · 2008
This work deals with the generalized eigenvalue problem for nonsquare matrix pencils $\mathbf{A}-\lambda \mathbf{B}$ such that matrices $\mathbf{A},\mathbf{B} \in \mathcal{M}_\mathbb{C} (m\times n)$ show a given structure. More precisely, we assume they result from removing the first row of some matrix $\mathbf{G} \in \mathcal{M}_\mathbb{C} ((m + 1),n)$ in the case of $\mathbf{A}$, and its last row in the case of $\mathbf{B}$. This structured generalized eigenvalue problem can be found in signal processing methods and in the numerical computation of the greatest common divisor (GCD) of polynomials. Traditional methods for solving the problem $(\mathbf{A}-\lambda \mathbf{A})\vect{v}=\vect{0}$ do not yield valid solutions when the data are not exact, as is often the case in real applications. In this work we adopt a minimal perturbation approach. Taking into account the structure of the matrices involved, we develop a simple algorithm for the computation of the generalized eigenvalues.