Drazin inverse

A Drazin inverse of an operator A is an operator, B, such that


where the spectral radiusMathworldPlanetmath r(U)=0. The Drazin inverse (B) is denoted by AD. It exists, if 0 is not an accumulation pointPlanetmathPlanetmath of σ(A).

For example, a projection operator is its own Drazin inverse, PD=P, as PPP=PP=P; for a Shift operator SD=0 holds.

The following are some other useful properties of the Drazin inverse:

  1. 1.


  2. 2.

    AD=(A+αP(A))-1(I-P(A)), where P(A):=I-ADA is the spectral projectionPlanetmathPlanetmath of A at 0 and α0;

  3. 3.

    A=(A*A)DA*=A*(AA*)D, where A is the Moore-Penrose pseudoinverseMathworldPlanetmathPlanetmath of A;

  4. 4.

    AD=Am(A2m+1)Am for mind(A), if ind(A):=min{k:ImAk=ImAk+1} is finite;

  5. 5.

    If the matrix is represented explicitly by its Jordan canonical formMathworldPlanetmath, (Λ regularPlanetmathPlanetmathPlanetmathPlanetmath and N nilpotentPlanetmathPlanetmath), then

  6. 6.

    Let eλA denote an eigenvectorMathworldPlanetmathPlanetmathPlanetmath of A to the eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath λ. Then eλA+t(λI-A)DheλA+O(t2) is an eigenvector of A+th.

Title Drazin inverse
Canonical name DrazinInverse
Date of creation 2013-03-22 13:58:05
Last modified on 2013-03-22 13:58:05
Owner kronos (12218)
Last modified by kronos (12218)
Numerical id 29
Author kronos (12218)
Entry type Definition
Classification msc 47S99
Related topic MoorePenroseGeneralizedInverse