Boley, Daniel Estimating the sensitivity of the algebraic structure of pencils with simple eigenvalue estimates. (English) Zbl 0725.15013 SIAM J. Matrix Anal. Appl. 11, No. 4, 632-643 (1990). 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. Reviewer: Min-Yen Wu (Boulder) Cited in 1 ReviewCited in 8 Documents MSC: 15A22 Matrix pencils 15A42 Inequalities involving eigenvalues and eigenvectors 93B05 Controllability 15A21 Canonical forms, reductions, classification 65F30 Other matrix algorithms (MSC2010) Keywords:Matrix pencils; controllability; linear time-invariant control systems; transmission zeros; differential algebraic equations; perturbations; Bauer-Fike theorem; eigenvalues; numerical examples; time-invariant PDFBibTeX XMLCite \textit{D. Boley}, SIAM J. Matrix Anal. Appl. 11, No. 4, 632--643 (1990; Zbl 0725.15013) Full Text: DOI