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 1 is called the group inverse or {1,2,5}-inverse and denoted A^#.

A projection P, as P² = P, has index 1 and 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 A₀: = αA or any regular A₀ with A₀A = AA₀ chosen such that $\|A_0-A_0 A A_0\|<\|A_0\|$ the sequence tends to its Drazin inverse,

$A_i \rightarrow A^D.$

See also

• Constrained generalized inverse
• Inverse element
• Moore–Penrose inverse

References

• Drazin, M. P., Pseudo-inverses in associative rings and semigroups, The American Mathematical Monthly 65(1958)506-514 JSTOR

• Bing Zheng and R. B. Bapat, Generalized inverse A(2)T,S and a rank equation, Applied Mathematics and Computation 155 (2004) 407-415 DOI 10.1016/S0096-3003(03)00786-0

External links

• Drazin inverse on Planet Math

This algebra-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Matrices Algebra stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем сделать НИР Juan Andrés Coloma Karl Gustav von Löwenwolde Look at other dictionaries: Drazin — may refer to: People Israel Drazin, US Brigadier General Michael P. Drazin, an American mathematician Drazin inverse, mathematical theory by Michael P. Drazin Places Dražin Do, a village in Bosnia and Herzegovina See also Draza (disambiguation)… …   Wikipedia Inverse element — In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can undo the effect of combination with… …   Wikipedia Moore-Penrose-Inverse — Die Pseudoinverse einer Matrix ist ein Begriff aus dem mathematischen Teilgebiet lineare Algebra. Sie ist eine Verallgemeinerung der inversen Matrix auf singuläre und nichtquadratische Matrizen, weshalb sie häufig auch als verallgemeinerte… …   Deutsch Wikipedia Michael P. Drazin — Michael Drazin Alma mater University of Cambridge Doctoral advisor Robert Rankin and David Rees …   Wikipedia Generalized inverse — In mathematics, a generalized inverse or pseudoinverse of a matrix A is a matrix that has some properties of the inverse matrix of A but not necessarily all of them. The term the pseudoinverse commonly means the Moore Penrose pseudoinverse. The… …   Wikipedia Pseudo-inverse — Pour les articles homonymes, voir Inverse (homonymie). En mathématiques, et plus précisément en algèbre linéaire, la notion de pseudo inverse (ou inverse généralisé) généralise celle d’inverse d’une application linéaire ou d’une matrice[1] aux… …   Wikipédia en Français Pseudoinverse — Die Pseudoinverse einer Matrix ist ein Begriff aus dem mathematischen Teilgebiet lineare Algebra. Sie ist eine Verallgemeinerung der inversen Matrix auf singuläre und nichtquadratische Matrizen, weshalb sie häufig auch als verallgemeinerte… …   Deutsch Wikipedia Moore–Penrose pseudoinverse — In mathematics, and in particular linear algebra, a pseudoinverse A+ of a matrix A is a generalization of the inverse matrix.[1] The most widely known type of matrix pseudoinverse is the Moore–Penrose pseudoinverse, which was independently… …   Wikipedia Optimal projection equations — In control, the Linear Quadratic Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive white Gaussian noise, incomplete state information (i.e. not all… …   Wikipedia List of mathematics articles (D) — NOTOC D D distribution D module D D Agostino s K squared test D Alembert Euler condition D Alembert operator D Alembert s formula D Alembert s paradox D Alembert s principle Dagger category Dagger compact category Dagger symmetric monoidal… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”