Moore-Penrose inverse of a matrix is the most widely known generalisation of the inverse matrix.
Note that the term 'generalised inverse' is sometimes used as a synonym for pseudopinverse.
History
1903 | Erik Ivar Fredholm introduced the concept of a pseudoinverse of integral operators |
1920 | The Moore-Penrose inverse was described by E. H. Moore |
1951 | The Moore-Penrose inverse was described by Arne Bjerhammar |
1955 | The Moore-Penrose inverse was described by Roger Penrose |
The common uses of the pseudoinverse are
- to compute a best-fit least squares solution to a system of linear equations that lacks a solution
- to find the minimum Euclidean norm solution to a system of linear equations with multiple solutions
The Moore-Penrose pseudoinverse is defined for any matrix and is unique.
The Moore-Penrose pseudoinverse P+ of a matrix P satisfies the following properties:
When P has linearly independent columns, P+ can be computed as
This pseudoinverse is a left inverse that is:
Since P+ is injective, the multiplication of P+ and P is invertible.
- Note that 'injective' means that a function maps distinct elements to distinct outputs. no two elements of P+ map to the same element of P.
When P has linearly independent rows, P+ can be computed as
This is a right inverse as:
Since P+ is surjective, the multiplication of P and P+ is invertible.
- Note that 'surjective' means that a function maps every element of its domain to an element of its codomain. A surjective matrix maps every element of the column space of the matrix to an element of the space.
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