On generalized inverses of singular matrix pencils

Abstract

Linear time-invariant networks are modelled by linear differential-algebraic equations with constant coefficients. These equations can be represented by a matrix pencil. Many publications on this subject are restricted to regular matrix pencils. In particular, the influence of the Weierstrass structure of a regular pencil on the poles of its inverse is well known. In this paper we investigate singular matrix pencils. The relations between the Kronecker structure of a singular matrix pencil and the multiplicity of poles at zero of the Moore-Penrose inverse and the Drazin inverse of the rational matrix are investigated. We present example networks whose circuit equations yield singular matrix pencils.

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“…The LS Prony method leads to high estimation performances compared to those of the Prony method. Other methods that lead to high performances under specific conditions are proposed in [83][84][85]. The Prony method allows for estimating inter-harmonics, harmonics and damping components.…”

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“…Occasionally, (general) pseudo-inverses are defined such that AA + A = A and A + AA + = A + hold, which ensures pseudo-invertibility of (some) singular matrices (see the works of Ben-Israel and Greville (2003), Bose and Mitra (1978), Boullion and Odell (1971), Campbell and Meyer (2008) or Röbenack and Reinschke (2011) in the context of dynamical systems). In addition to the above conditions, the Moore-Penrose pseudo-inverse satisfies (A + A) T = A + A and (AA + ) T = AA + , and although it can be used here, we can mostly be more permissive for the purpose of this paper.…”

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