Drazin inverse

In mathematics, the Drazin inverse, named after Michael P. Drazin, is a kind of generalized inverse of a matrix.
Let "A" be a square matrix. The index of "A" is the least nonnegative integer "k" such that rank("A""k"+1) = rank("A""k"). The Drazin inverse of "A" is the unique matrix "A"D which satisfies
It's not a generalized inverse in the classical sense, since formula_2 in general.
The hyper-power sequence is 
For formula_12 or any regular formula_13 with formula_14 chosen such that formula_15 the sequence tends to its Drazin inverse, 
= Jordan normal form =
As the definition of the Drazin inverse is invariant under matrix conjugations, writing formula_17 where J is in Jordan normal form, implies that formula_18. The Drazin inverse is then the operation that maps invertible Jordan blocks to their inverses, and nilpotent Jordan blocks to zero.