Pseudo-inverse of A
Here's what happens if some of the singular values are zero.
Suppose where and are orthonormal bases for and and are non-zero.
Carry out the three steps of solving using a matrix to carry out each step.
Step 1: Find coords of .
Since is an orthonormal basis the coords are .
But since you aren't interested in .
Choose . Then .
Step 2: Divide the numbers by the singular values.
Step 3: Use the entries of as the coords of .
Combining all three steps, a least squares solution to is
The matrix product is the pseudo-inverse of .
You get the pseudo-inverse by keeping the parts of that correspond to non-zero singular values.
Computing by hand is a hard job so you usually use a machine. In Mathematica you use PseudoInverse[A] and the only hard thing is remembering how to spell pseudo.
One big advantage of using to solve is that you don't need to know in advance whether has solutions.
Without any worries you compute .
If has solutions, will be one of them.
If has NO solutions, then will give you a least squares solution.
(Q44) Suppose .
Express as a product of three matrices.
Copyright © 2007 Todd Will
Created by Mathematica (April 15, 2007)