MATH347DS L09: SVD computations, pseudo-inverse, model reduction

Overview

• SVD visualization

• Computing the SVD

SVD visualization

• Consider $𝑨\in {ℝ}^{2\times 2}$

• Obtain above figure by:

  • Place $𝒙$ on the unit circle: $𝒙={\left[\begin{array}{ll}\cos \theta & \sin \theta \end{array}\right]}^T$

  • Compute $𝒚=𝑨𝒙={\left[\begin{array}{ll}2\cos \theta -\sin \theta & 3\cos \theta +\sin \theta \end{array}\right]}^T$

  • Plot $𝒙\left(\theta \right)$, $𝒚\left(\theta \right)$

SVD computation

• From $𝑨=𝑼𝚺{𝑽}^T$ deduce $𝑨{𝑨}^T=𝑼{𝚺}^2{𝑼}^T$, $𝑨{}^T𝑨=𝑽{𝚺}^2{𝑽}^T$, hence $𝑼$ is the eigenvector matrix of $𝑨{𝑨}^T$, and $𝑽$ is the eigenvector matrix of ${𝑨}^T𝑨$

• SVD computation is carried out by solving eigenvalue problems

octave> 

A=[2 -1; -3 1]; [U S2]=eig(A*A'); [V S2]=eig(A'*A); S=sqrt(S2); disp([ U S V']);

• The Octave/Matlab svd function carries out both eigenvalue problems efficiently

octave> 

[U S V]=svd(A); disp([U S V']);

octave> 

disp([U*U' U'*U V*V' V'*V]);

octave>