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 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 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.
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A generalized inverse (denoted by a superscript of -) of a rectangular matrix of values A is any matrix that satisfies
A-AA=A
A generalized inverse of a nonsingular matrix is unique and is called the regular matrix inverse.
See also matrix singularity , matrix inverse .

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