Drazin inverse

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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 AD which satisfies

  • If A is invertible with inverse , then .
  • The Drazin inverse of a matrix of index 0 or 1 is called the group inverse or {1,2,5}-inverse and denoted A#. The group inverse can be defined, equivalently, by the properties AA#A = A, A#AA# = A#, and AA# = A#A.
  • A projection matrix P, defined as a matrix such that P2 = P, has index 1 (or 0) and has Drazin inverse PD = P.
  • If A is a nilpotent matrix (for example a shift matrix), then

The hyper-power sequence is

for convergence notice that

For or any regular with chosen such that the sequence tends to its Drazin inverse,

See also[edit]


  • Drazin, M. P., (1958). "Pseudo-inverses in associative rings and semigroups". The American Mathematical Monthly. 65 (7): 506–514. doi:10.2307/2308576. JSTOR 2308576.
  • Zheng, Bing; Bapat, R.B (2004). "Generalized inverse A(2)T,S and a rank equation". Applied Mathematics and Computation. 155 (2): 407. doi:10.1016/S0096-3003(03)00786-0.

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