Drazin Inverse
   HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 inv ...
of a
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
. Let ''A'' be a square matrix. The
index Index (: indexes or 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 the Halo Array in the ...
of ''A'' is the least nonnegative integer ''k'' such that
rank A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy ...
(''A''''k''+1) =
rank A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy ...
(''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 inv ...
in the classical sense, since A A^\text A \neq A in general. * If ''A'' is invertible with inverse A^, then A^\text = A^. * If ''A'' is a block diagonal matrix :A = \begin B & 0 \\ 0 & N \end where B is invertible with inverse B^ and N 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 trans ...
, then :A^D = \begin B^ & 0 \\ 0 & 0 \end * 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 trans ...
(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.


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


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 no multiple roots in any field extension ''F/k''. * Every irreducible polynomial over ''k'' has non-zero f ...
, 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 , often called the pseudoinverse, is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Ar ...
*
Jordan normal form \begin \lambda_1 1\hphantom\hphantom\\ \hphantom\lambda_1 1\hphantom\\ \hphantom\lambda_1\hphantom\\ \hphantom\lambda_2 1\hphantom\hphantom\\ \hphantom\hphantom\lambda_2\hphantom\\ \hphantom\lambda_3\hphantom\\ \hphantom\ddots\hphantom\\ ...
*
Generalized eigenvector In linear algebra, a generalized eigenvector of an n\times n matrix A is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector. Let V be an n-dimensional vector space and let A be the matrix r ...


References

* * *


External links


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