Pseudo inverse matrix

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Property 1

Suppose that A is mn real matrix

If m<n, then the inverse of ATA does not exist

If mn and if the inverse of ATA exists

A+=(ATA)-1AT satisfies the definition of pseudoinverse

Here, A+A=I holds. I is identity matrix.

A: mn, AT: nm, A+: nm, ATA: 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 AAT does not exist

If mn and if the inverse of AAT exists

A+=AT(AAT)-1 satisfies the definition of pseudoinverse

Here, AA+=I holds. I is identity matrix.

A: mn, AT: nm, A+: nm, AAT: 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-1A=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, ATA and AAT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=UWVT 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(w1,w2,...,wk,0,0,...,0), we define

W''=diag(1/w1,1/w2,...,1/wk,0,0,...,0)

Define A+=VW''UT

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''UTb

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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