In mathematics, the Drazin inverse, named after Michael P. Drazin, is a kind of generalized inverse of a matrix.
The index of A is the least nonnegative integer k such that rank(Ak+1) = rank(Ak).
The Drazin inverse of A is the unique matrix AD that satisfies It's not a generalized inverse in the classical sense, since
is a nilpotent matrix, then The hyper-power sequence is For
the sequence tends to its Drazin inverse, A study of Drazin inverses via category-theoretic techniques, and a notion of Drazin inverse for a morphism of a category, has been recently initiated by Cockett, Pacaud Lemay and Srinivasan.
This notion is a generalization of the linear algebraic one, as there is a suitably defined category
having morphisms matrices
with complex entries; a Drazin inverse for the matrix M amounts to a Drazin inverse for the corresponding morphism in
As the definition of the Drazin inverse is invariant under matrix conjugations, writing
, where J is in Jordan normal form, implies that
The Drazin inverse is then the operation that maps invertible Jordan blocks to their inverses, and nilpotent Jordan blocks to zero.
More generally, we may define the Drazin inverse over any perfect field, by using the Jordan-Chevalley decomposition
is nilpotent and both operators commute.
The two terms can be block diagonalized with blocks corresponding to the kernel and cokernel of
The Drazin inverse in the same basis is then defined to be zero on the kernel of
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