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In mathematics, the Drazin inverse, named after Michael P. Drazin, is a kind of
generalized inverse In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized in ...
of a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
. Let ''A'' be a square matrix. The
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
of ''A'' is the least nonnegative integer ''k'' such that
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...
(''A''''k''+1) =
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...
(''A''''k''). The Drazin inverse of ''A'' is the unique matrix ''A''D that satisfies :A^A^\text = A^k,\quad A^\textAA^\text = A^\text,\quad AA^\text = A^\textA. It's not a
generalized inverse In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized in ...
in the classical sense, since A A^\text A \neq A in general. * If ''A'' is invertible with inverse A^, then A^\text = A^. * Drazin inversion is invariant under conjugation. If A^\text is the Drazin inverse of A, then P A^\text P^ is the Drazin inverse of PAP^. * 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 In statistics, the projection matrix (\mathbf), sometimes also called the influence matrix or hat matrix (\mathbf), maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). It describes ...
''P'', defined as a matrix such that ''P''2 = ''P'', has index 1 (or 0) and has Drazin inverse ''P''D = ''P''. * If A is a
nilpotent matrix In linear algebra, a nilpotent matrix is a square matrix ''N'' such that :N^k = 0\, for some positive integer k. The smallest such k is called the index of N, sometimes the degree of N. More generally, a nilpotent transformation is a linear tran ...
(for example a shift matrix), then A^\text = 0. The hyper-power sequence is :A_ := A_i + A_i\left(I - A A_i\right); for convergence notice that A_ = A_i \sum_^ \left(I - A A_i\right)^k. 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 \rightarrow A^\text.


Jordan normal form and Jordan-Chevalley decomposition

As the definition of the Drazin inverse is invariant under matrix conjugations, writing A = P J P^, where J is in Jordan normal form, implies that A^\text = P J^\text P^ . The Drazin inverse is then the operation that maps invertible Jordan blocks to their inverses, and nilpotent Jordan blocks to zero. More generally, we may define the Drazin inverse over any
perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k' ...
, by using the Jordan-Chevalley decomposition 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 In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
*
Moore–Penrose inverse In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Roge ...
*
Jordan normal form In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to ...
* Generalized eigenvector


References

* *


External links


Drazin inverse
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Planet MathGroup inverse
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Planet Math
Matrices {{Linear-algebra-stub de:Pseudoinverse#Ausgewählte weitere Versionen von verallgemeinerten Inversen