SVD

description

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• similar to spectral factorization
• any non-zero matrix $A\in R^{m,n}$, can be factored as $A=U\tilde{S}{V}^T$
  • $U$: orthogonal matrix, $\in R^{m,m}$. columns of U: left-singular vectors
  • $\tilde{S}$: diagonal, positive & decreasing in magnitude entries
  • $V$: orthogonal matrix, $\in R^{n,n}$. columns of V: right-singular vectorsf

finding SVD of matrix C

? 0. want to find $C=US{V}^T$

1. Knowing that $C^{T}C=V{S}^T{U}^{T}US{V}^T=V{S}^{T}S{V}^T$, find $C^{T}C$ → find $V$ and $S$
   • find $C^{T}C$
   • find $det(C^{T}C-\lambda I)$ = 0 (finding eigenvectors AND eigenvalues)
2. From $CV=U.S$ → find $U=C.V.S^T$

compact form SVD

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• any non-zero matrix $A\in R^{m,n}$ can be expressed as: $A=U_r.S.V_r^T$

SVD from dyads view (lec 19)

• any matrix $A\in R^{m,n}$, rank $r$ > 0 can be expressed as sum of “orthogonal dyads”
• $A={\sum }_{i=1}^r{\sigma }_i{u}_i{v}_i^{\top }$
  • $u_1,\cdots ,u_r\in R^m$ , $v_1,\cdots ,v_r\in R^n$ are mutually orthogonal collections of vectors
  • ${\sigma }_1\ge \cdots \ge {\sigma }_r>0$
  • now completing these collections with vectors $v_i\in R^m,i=1,\cdots ,n$ and $u_i\in R^m,i=r+1,\cdots ,m$, so that $U=u_1,\cdots ,u_m$ and $V=[v_1,\cdots ,v_n$ ] are both orthonormal matrices