On the Segré, Weyr characteristics of right (left) regular matrix pencils

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Abstract
The classical Segré, Weyr characteristic theory of the standard eigenvalue-eigenvector problem defined on A ϵ ℝn×n is 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 Jα r(F,G) (Jα 1(F,G)) respectively are introduced. For right (left) regular pairs it is shown that Jα r(F,G) (Jα 1(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.