Computing Stable Elgendecompositions of Matrix Pencils (Classic Reprint)

Understand how small changes affect generalized eigenstructures in matrix pencils. This book explains a framework for computing stable decompositions of pencils A − λB, focusing on when the decomposition remains valid as the data varies within a tolerance. It links perturbation bounds to the block structure that reveals regular and singular parts, including left and right deflating subspaces.

In clear terms, the work introduces criteria to decide if a decomposition is stable, discusses the Kronecker Canonical Form, and shows how to bound how much the defining matrices P, Q, S, and T can change as A and B vary. It also connects theory to practical questions in controllability and observability, and describes how to verify stability using computable quantities.

• Shows how perturbations influence the spectrum and deflating subspaces of A − λB.


• Defines measurable bounds that determine when a decomposition remains valid across pencils in a family P(ε).


• Relates stability to condition numbers of the deflating subspaces and to the sizes of finite and infinite eigenvalue blocks.


• Applies to control system concepts like controllability and observability subspaces with a perturbation perspective.

Ideal for readers of advanced linear algebra and numerical analysis who want a rigorous treatment of stability in generalized eigenproblems and practical criteria for deciding when a decomposition is reliable.