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 which satisfies
It's not a generalized inverse in the classical sense, since in general.
- If A is invertible with inverse , 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,
Jordan normal form[]
As the definition of the Drazin inverse is invariant under matrix conjugations, writing where J is in Jordan normal form, implies that . The Drazin inverse is then the operation that maps invertible Jordan blocks to their inverses, and nilpotent Jordan blocks to zero.
See also[]
- Constrained generalized inverse
- Inverse element
- Moore–Penrose inverse
- Jordan normal form
- Generalized eigenvector
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[]
Categories:
- Matrices
- Linear algebra stubs