On the Canonical Forms of a Regular Matrix Pencil

2021 ◽  
Author(s):  
P. F. Samusenko
Keyword(s):  
Matrix Pencil ◽  
Canonical Forms ◽  
Regular Matrix

A new bound for the distance to singularity of a regular matrix pencil

PAMM ◽  
2017 ◽  
Vol 17 (1) ◽  
pp. 863-864 ◽  
Author(s):  
Thomas Berger ◽  
Hannes Gernandt ◽  
Carsten Trunk ◽  
Henrik Winkler ◽  
Michał Wojtylak
Keyword(s):  
Matrix Pencil ◽  
Regular Matrix

On Wilkinson's problem for matrix pencils

2015 ◽  
Vol 30 ◽  
pp. 632-648 ◽  
Author(s):  
Sk. Ahmad ◽  
Rafikul Alam
Keyword(s):  
Condition Number ◽  
Matrix Pencil ◽  
Upper Bounds ◽  
Regular Matrix ◽  
Eigenvalues And Eigenvectors ◽  
Lower And Upper Bounds ◽  
Matrix Pencils

Suppose that an n-by-n regular matrix pencil A -\lambda B has n distinct eigenvalues. Then determining a defective pencil E−\lambda F which is nearest to A−\lambda B is widely known as Wilkinson’s problem. It is shown that the pencil E −\lambda F can be constructed from eigenvalues and eigenvectors of A −\lambda B when A − \lambda B is unitarily equivalent to a diagonal pencil. Further, in such a case, it is proved that the distance from A −\lambda B to E − \lambdaF is the minimum “gap” between the eigenvalues of A − \lambdaB. As a consequence, lower and upper bounds for the “Wilkinson distance” d(L) from a regular pencil L(\lambda) with distinct eigenvalues to the nearest non-diagonalizable pencil are derived.Furthermore, it is shown that d(L) is almost inversely proportional to the condition number of the most ill-conditioned eigenvalue of L(\lambda).

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