On the Segré, Weyr characteristics of right (left) regular matrix pencils
作者:
NICOS KARCANIAS,
GRIGORIS KALOGEROPOULOS,
期刊:
International Journal of Control
(Taylor Available online 1986)
卷期:
Volume 44,
issue 4
页码: 991-1015
ISSN:0020-7179
年代: 1986
DOI:10.1080/00207178608933647
出版商: Taylor & Francis Group
数据来源: Taylor
摘要:
The classical Segré, Weyr characteristic theory of the standard eigenvalue-eigenvector problem defined on A ϵ ℝn×nis extended to the case of right (left) regular pencils sF — G, F, G ϵ ℝm×n. The notions of α-Toeplitz matrices and of the right (left) α. — (F, G) sequences Jαr(F,G) (Jα1(F,G)) respectively are introduced. For right (left) regular pairs it is shown that Jαr(F,G) (Jα1(F,G)) are piecewise arithmetic progression sequences (PAPS); the singular points of those PAPS define the possible degrees of the elementary divisors (e.d.) at α, whereas the deviations, or gaps from the arithmetic progression sequence, define the multiplicity of the corresponding degree e.d.s. The links between the algebraic notion of the Segré characteristic and the geometric notion of the Weyr characteristic are established for such families of pencils. Two different procedures for the computation of the Segré characteristic are given; the first is based on the analysis of singular points of PAPS and the second is based on the conjugate partitioning property of the algebraic multiplicity of a zero of sF - G at α by the Weyr and Segré characteristics at a. In the latter case a Ferrers-type diagram is suggested as the tool for the computation of the Segré characteristic from the Weyr characteristic. The results provide a procedure for the computation of the Weierstrass part of the Kronecker form of right (left) regular pencils by rank tests on α-Toeplitz matrices.
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