Generalized inverse
Encyclopedia
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 purpose of constructing a generalized inverse is to obtain a matrix that can serve as the inverse in some sense for a wider class of matrices than invertible ones. Typically, the generalized inverse exists for an arbitrary matrix, and when a matrix has an inverse, then its inverse and the generalized inverse are the same. Some generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a generalized inverse or pseudoinverse of a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
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 purpose of constructing a generalized inverse is to obtain a matrix that can serve as the inverse in some sense for a wider class of matrices than invertible ones. Typically, the generalized inverse exists for an arbitrary matrix, and when a matrix has an inverse, then its inverse and the generalized inverse are the same. Some generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
.
Types of generalized inverses
The various kinds of generalized inverses include- One-sided inverse (left inverse or right inverse) If the matrix A has dimensions
then use the left inverse if 
and the right inverse if 
- Left inverse is given by
, i.e. 
where 
is the 
identity matrixIdentity matrixIn linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
. - Right inverse is given by
, i.e. 
where 
is the 
identity matrix.
- Left inverse is given by
- Drazin inverseDrazin inverseIn 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 = rank...
- Bott–Duffin inverse
- Moore–Penrose pseudoinverse
External links
- 15A09 Matrix inversion, generalized inverses in Mathematics Subject ClassificationMathematics Subject ClassificationThe Mathematics Subject Classification is an alphanumerical classification scheme collaboratively produced by staff of and based on the coverage of the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH...
, MathSciNet search