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(Ak+1) = rank(Ak). The Drazin inverse of A is the unique matrix AD that satisfies
It's not a generalized inverse in the classical sense, since in general.
- If A is invertible with inverse , then .
- If A is a block diagonal matrix
where is invertible with inverse and is a nilpotent matrix, then
- Drazin inversion is invariant under conjugation. If is the Drazin inverse of , then is the Drazin inverse of .
- 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,