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Drazin Inverse

The Drazin inverse is a matrix inverse-like object derived from a given square matrix. In particular, let the index k of a square matrix be defined as the smallest nonnegative integer such that the matrix rank satisfies rank(A^(k+1))=rank(A^k). Then the Drazin inverse is the unique matrix A^D such that

A^(k+1)A^D=A^k
(1)
A^DAA^D=A^D
(2)
AA^D=A^DA.
(3)

If A is an invertible matrix with matrix inverse A^(-1), then A^D=A^(-1).

The Drazin inverse is implemented in the Wolfram Language as DrazinInverse[m].

See also

Matrix 1-Inverse, Matrix Inverse Moore-Penrose Matrix Inverse, Pseudoinverse

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References

Drazin, M. P. "Pseudo-Inverses in Associative Rings and Semigroups." Amer. Math. Monthly 65, 506-514, 1958.

Cite this as:

Weisstein, Eric W. "Drazin Inverse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DrazinInverse.html

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