Pseudo Inverse Matrix

Matriks yang tidak persegi membutuhkan suatu metode khusus untuk menginverskan, yang biasa disebut pseudo inverse matriks.

Properti 1
Suppose that A is mn real matrix
If m<n, then the inverse of A^TA does not exist
If mn and if the inverse of A^TA exists
A⁺=(A^TA)^-1A^T satisfies the definition of pseudoinverse
Here, A⁺A=I holds. I is identity matrix.
A: mn, A^T: nm, A⁺: nm, A^TA: nn, I: nn
The rank of A and A⁺ is n

Properti 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

Pseudoinverse dengan Teknik Singular Decomposition Value (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(w₁,w₂,...,w_k,0,0,...,0), we define
W''=diag(1/w₁,1/w₂,...,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^Tb
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


Reference : Diktat Mata Kuliah Teknik Numerik Sistem Linier, ITS Surabaya