Moore-Penrose inverse 무어-펜로즈 의사역행렬

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

1. to compute a best-fit least squares solution to a system of linear equations that lacks a solution
2. 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.

https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection