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Estimating the sensitivity of the algebraic structure of pencils with simple eigenvalue estimates. (English) Zbl 0725.15013

Matrix pencils have many interesting applications in the controllability theory of linear time-invariant control systems, in the theory of transmission zeros and in the theory of differential algebraic equations. The author studies the sensitivity of the algebraic structure of rectangular matrix pencils to perturbations in the coefficients. He first examines a method for checking whether or not a given pencil is deficient. The modified Bauer-Fike theorem, which gives bounds on the changes of the eigenvalues under perturbations, is then used to develop upper bounds and lower bounds for the distance from a given non-deficient pencil to the nearest deficient pencil. It is also noted that the scheme given in the paper for estimating the distance from a given pencil to the nearest pencil of different Kronecker structure is less sensitive to the particular choice of zero tolerance. Some numerical examples are also given for comparing results obtained from using different schemes.
This paper is very well presented.

MSC:

15A22 Matrix pencils
15A42 Inequalities involving eigenvalues and eigenvectors
93B05 Controllability
15A21 Canonical forms, reductions, classification
65F30 Other matrix algorithms (MSC2010)
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