Core-Nilpotent Decomposition

From Engineering Math

Definition[edit]

If is an matrix such that , then there exists a nonsungular matrix such that (p396 [1])

 

in which is nonsingular and is nilpotent of index .

In other words, is similar to a block-diagonal matrix containing a non-singular core and a nilpotent component.

The Drazin Inverse[edit]

Drazin is the quintessential pseudoinverse for the core-nilpotent decomposition, defined as:


Note that Drazin is not a reflexive pseudoinverse, but an outer pseudoinverse. That is,


References[edit]

  1. Carl D. Meyer, "Matrix Analysis and Applied Linear Algebra"