Drazin inverse

From Wikipedia, the free encyclopedia

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(Ak+1)=rank(Ak). The Drazin inverse of A is the unique matrix X, which satisfies

A^{k+1}X=A^k, \quad XAX=X,\quad AX=XA.

The Drazin inverse of a matrix of index 1 is called the group inverse.

[edit] References

  • Drazin, M. P., Pseudo-inverses in associative rings and semigroups, The American Mathematical Monthly 65(1958)506-514 JSTOR
  • Bing Zheng and R. B. Bapat, Generalized inverse A(2)T,S and a rank equation, Applied Mathematics and Computation 155 (2004) 407-415 DOI 10.1016/S0096-3003(03)00786-0

[edit] External links

Drazin inverse on Planet Math

This algebra-related article is a stub. You can help Wikipedia by expanding it.