Drazin inverse explained

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 that satisfies

A ^{k+1} A ^{D} = A ^{k,} A ^{DAA} ^{D} = A ^{D,} AA ^{D} = A ^{DA.}

It's not a generalized inverse in the classical sense, since

A A ^{D} A ≠ A in general.A ^{-1} , then A ^{D} = A ^{-1} .

• If A is a block diagonal matrix

A = \begin{bmatrix} B & 0 \\ 0 & N \end{bmatrix} where B is invertible with inverse B ^{-1} and N is a nilpotent matrix, then A ^{D} = \begin{bmatrix} B ^{-1} & 0 \\ 0 & 0 \end{bmatrix}

• Drazin inversion is invariant under conjugation. If

A ^{D} is the Drazin inverse of A , then P A ^{D} P ^{-1} is the Drazin inverse of PAP ^{-1} .

• The Drazin inverse of a matrix of index 0 or 1 is called the group inverse or -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 ^{2 ^{j-1}} _{k =0} \left(I - A A ^{k.} _{i\right)} For A _{0} : = \alpha A or any regular A _{0} with A _{0} A = A A _{0} chosen such that \left\|A _{0} - A _{0} A A _{0\right\|} < \left\|A _{0\right\|} the sequence tends to its Drazin inverse, A _{i} → A ^{D.}

Drazin inverses in categories

A study of Drazin inverses via category-theoretic techniques, and a notion of Drazin inverse for a morphism of a category, has been recently initiated by Cockett, Pacaud Lemay and Srinivasan. This notion is a generalization of the linear algebraic one, as there is a suitably defined category

MAT having morphisms matrices M : C ^{n\to} C ^{m} with complex entries; a Drazin inverse for the matrix M amounts to a Drazin inverse for the corresponding morphism in MAT .

Jordan normal form and Jordan-Chevalley decomposition

As the definition of the Drazin inverse is invariant under matrix conjugations, writing

A = P J P ^{-1} , where J is in Jordan normal form, implies that A ^{D} = P J ^{D} P ^{-1} . The Drazin inverse is then the operation that maps invertible Jordan blocks to their inverses, and nilpotent Jordan blocks to zero.A = A _{s} + A _{n} where A _{s} is semisimple and A _{n} is nilpotent and both operators commute. The two terms can be block diagonalized with blocks corresponding to the kernel and cokernel of A _{s} . The Drazin inverse in the same basis is then defined to be zero on the kernel of A _{s} , and equal to the inverse of A on the cokernel of A _{s} .

See also

• Constrained generalized inverse
• Inverse element
• Moore–Penrose inverse
• Jordan normal form
• Generalized eigenvector

References

• Drazin. M. P.. Pseudo-inverses in associative rings and semigroups. The American Mathematical Monthly. 65. 1958. 506–514. 2308576. 10.2307/2308576. 7.
• 10.1016/S0096-3003(03)00786-0 . Generalized inverse A(2)T,S and a rank equation. 2004. Zheng. Bing. Bapat. R.B. Applied Mathematics and Computation. 155. 2. 407.
• Drazin Inverses in Categories . 2024 . Cockett . Robin . Pacaud Lemay . Jean-Simon . Srinivasan . Priyaa Varshinee . 2402.18226 . math.CT.

External links

• Drazin inverse on Planet Math
• Group inverse on Planet Math