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

A^{k+1}A^{\text{D}}=A^{k},\quad A^{\text{D}}AA^{\text{D}}=A^{\text{D}},\quad AA^{\text{D}}=A^{\text{D}}A.

It's not a generalized inverse in the classical sense, since $AA^{\text{D}}A\neq A$ in general.

If A is invertible with inverse $A^{-1}$ , then $A^{\text{D}}=A^{-1}$ .
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 P^D = P.
If A is a nilpotent matrix (for example a shift matrix), then $A^{D}=0.$

The hyper-power sequence is

A_{i+1}:=A_{i}+A_{i}\left(I-AA_{i}\right);

for convergence notice that

A_{i+j}=A_{i}\sum _{k=0}^{2^{j}-1}\left(I-AA_{i}\right)^{k}.

For $A_{0}:=\alpha A$ or any regular $A_{0}$ with $A_{0}A=AA_{0}$ chosen such that $\left\|A_{0}-A_{0}AA_{0}\right\|<\left\|A_{0}\right\|$ the sequence tends to its Drazin inverse,

A_{i}\rightarrow A^{\text{D}}.

References

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.

External links

Drazin inverse on Planet Math
Group inverse on Planet Math

Drazin inverse

See also

References

External links