8.5 Singular Value Decomposition

What is SVD

• Recall that A is orthogonally diagonalizable if there exists an orthogonal matrix P and diagonal matrix D

  • $P^{T}AP=D$ this is just 6.2 Diagonalization
  • what if we allow orthogonal matrices U and V to be used in “diagonalization” instead of P
    • $A=U\Sigma V^T$

How to find SVD

• To find the values of V and $\Sigma$

  • we must do $A^{T}A$
    • this simplifies to $V{\Sigma }^T\Sigma V^T$
      • since ${\Sigma }^T\Sigma ={\Sigma }^2$ is a diagonal matrix, this tells us that $A^{T}A$ needs to be orthogonally diagonalized with
        • $P=V,D={\Sigma }^2$