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^D=A^k, \quad A^D A A^D=A^D,\quad A A^D= A^D A.

If A is invertible with inverse $A^{-1}$ , then $A^D=A^{-1}$ .

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- A A_i\right);

for convergence notice that

A_{i+j}=A_i \sum_{k=0}^{2^j-1} (I-A A_i)^k.

For $A_0:=\alpha A$ or any regular $A_0$ with $A_0 A= A A_0$ chosen such that $\|A_0-A_0 A A_0\|<\|A_0\|$ the sequence tends to its Drazin inverse,

A_i \rightarrow A^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