In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Drazin inverse, named after
Michael P. Drazin
Michael Peter Drazin (born 1929) is an American mathematician of British background, working in noncommutative algebra.
Background
The Drazins (Дразин) were a Russian Jewish family who moved to the United Kingdom in the years before World ...
, 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 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'' (franchis ...
.
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
:
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
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 -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 t ...
''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 transf ...
(for example a
shift matrix In mathematics, a shift matrix is a binary matrix with ones only on the superdiagonal or subdiagonal, and zeroes elsewhere. A shift matrix ''U'' with ones on the superdiagonal is an upper shift matrix. The alternative subdiagonal matrix ''L'' is ...
), 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 and Jordan-Chevalley decomposition
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.
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'' is ...
, by using the
Jordan-Chevalley decomposition where
is semisimple and
is nilpotent and both operators commute. The two terms can be block diagonalized with blocks corresponding to the kernel and cokernel of
. The Drazin inverse in the same basis is then defined to be zero on the kernel of
, and equal to the inverse of
on the cokernel of
.
See also
*
Constrained generalized inverse
In linear algebra, a constrained generalized inverse is obtained by solving a system of linear equations with an additional constraint that the solution is in a given subspace. One also says that the problem is described by a system of constraine ...
*
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 is ...
*
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 Roger Pe ...
*
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 som ...
*
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; let \phi be a linear map ...
References
*
*
External links
Drazin inverseo
Planet MathGroup inverseo
Planet Math
Matrices
{{Linear-algebra-stub
de:Pseudoinverse#Ausgewählte weitere Versionen von verallgemeinerten Inversen