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Preprint 12-2004

An Arithmetic for Matrix Pencils: Theory and New Algorithms

Source file is available as :   Portable Document Format (PDF)

Author(s) : Ralph Byers , Peter Benner

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 12-2004

MSC 2000

65F15 Eigenvalues, eigenvectors
65F30 Other matrix algorithms

Abstract :
This paper introduces arithmetic-like operations on matrix pencils. The pencil-arithmetic operations extend elementary formulas for sums and products of rational numbers and include the algebra of linear transformations as a special case. These operations give an unusual perspective on a variety of pencil related computations. We derive generalizations of monodromy matrices and the matrix exponential. A new algorithm for computing a pencil arithmetic generalization of the matrix sign function does not use matrix inverses and gives an empirically forward numerically stable algorithm for extracting deflating subspaces.

Keywords : matrix pencil, matrix sign function, differential algebraic equation

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