# Pseudo inverse matrix

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## Property 1

Suppose that A is mn real matrix

If m<n, then the inverse of A^{T}A does not exist

If mn and if the inverse of A^{T}A exists

A^{+}=(A^{T}A)^{-1}A^{T}
satisfies the definition of pseudoinverse

Here, A^{+}A=I holds. I is identity matrix.

A: mn, A^{T}:
nm, A^{+}:
nm, A^{T}A:
nn, I: nn

The rank of A and A^{+} is n

## Property 2

Suppose that A is mn real matrix

If m>n, then the inverse of AA^{T} does not exist

If mn
and if the inverse of AA^{T} exists

A^{+}=A^{T}(AA^{T})^{-1}
satisfies the definition of pseudoinverse

Here, AA^{+}=I holds. I is identity matrix.

A: mn, A^{T}:
nm, A^{+}:
nm, AA^{T}:
mm, I: mm

The rank of A and A^{+} is m

## Inverse

If A is square matrix, and if the inverse of A exists, then A^{+}=A^{-1}
holds. For the above two A^{+}, AA^{+}=A^{+}A=AA^{-1}=A^{-1}A=I
holds. I is identity matrix.

## Numerical computation

The methods like Gauss-Jordan or LU decomposition can only
calculate the inverse of square non-singular matrix. Even if A is
not square, A^{T}A and AA^{T}が become square;
thus, it may be possible to apply Gauss-Jordan or LU
decomposition. Singular Value Decomposition (SVD) may also be
used for calculating the pseudoinverse.

## Pseudoinverse by Singular Value Decomposition (SVD)

Suppose A is mn
matrix. If m<n, attach the row of 0 and make the size m=n, a
priori. Here, mn.

A=UWV^{T} is supposed to be the result of SVD

Assume that the left-upper part of W has larger number, and
the right-lower part of W has smaller number

If the component of W is less than a threshold, set it to be
0, and define such matrix as W'

When W'=diag(w_{1},w_{2},...,w_{k},0,0,...,0),
we define

W''=diag(1/w_{1},1/w_{2},...,1/w_{k},0,0,...,0)

Define A^{+}=VW''U^{T}

A^{+}A is nn matrix. Left-upper kk is identity matrix. Otherwise 0.

AA^{+} is mm matrix

If the rank of A is n, A^{+} satisfies the above
property 1

Even if the rank of A is less than n, A^{+} satisfies
the definition of pseudoinverse

Suppose that we want to solve Ax=b. Calculate x=VW''U^{T}b

If the rank of A is n, then x is the value where the error is
minimum

If the rank of A is less than n, then x is the solution where
the norm ||x|| is minimum

## Definition of Moore-Penrose generalized matrix inverse

Given mn real
matrix A, nm
matrix pseudoinverse A^{+} is defined as follows

AA^{+}A=A

A^{+}AA^{+}=A^{+}

(AA^{+})^{T}=AA^{+}

(A^{+}A)^{T}=A^{+}A

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