Pseudo-inverse of a matrix via its SVD

Returning to this example, the pseudo-inverse of the matrix

A = left(begin{array}{ccccc} 1 & 0 & 0 & 0 & 2 0 & 0 & 3 & 0 & 0 0 & 0 & 0 & 0 & 0 0 & 4 & 0 & 0 & 0 end{array}right).

Can be computed via an SVD: $A = U tilde{ {S}} V^T$ , with

$U = left(begin{array}{cccc}0 & 0 & 1 & 00 & 1 & 0 & 00 & 0 & 0 &-11 & 0 & 0 & 0 end{array}right) , ;; tilde{ {S}} = left(begin{array}{ccccc} 4 & 0 & 0 & 0 & 00 & 3 & 0 & 0 & 00 & 0 &sqrt{5} & 0 & 00 & 0 & 0 & 0 & 0 end{array}right) , ;; V^T = left(begin{array}{ccccc}0 & 1 & 0 & 0 & 0 0 & 0 & 1 & 0 & 0 sqrt{0.2} & 0 & 0 & 0 & sqrt{0.8}0 & 0 & 0 & 1 & 0-sqrt{0.8} & 0 & 0 & 0 &sqrt{0.2} end{array}right) ,$

as follows.

We first invert tilde{ {S}} , simply ‘‘inverting what can be inverted’’ and leaving zero values alone. We get

$tilde{ {S}}^dagger = left(begin{array}{ccccc} 1/4 & 0 & 0 & 0 0 & 1/3 & 0 & 0 0 & 0 &1/sqrt{5} & 0 0 & 0 & 0 & 0 0 & 0 & 0 & 0 end{array}right).$

Then the pseudo-inverse is obtained by exchanging the roles of U,V in the SVD:

$A^dagger = V tilde{ {S}}^dagger U^T = left(begin{array}{ccccc} 0.2000 & 0 & 0 & 0 0 & 0 & 0 & 0.2500 0 & 0.3333 & 0 & 0 0 & 0 & 0 & 0 0.4000 & 0 & 0 & 0 end{array}right).$

See also: this example.