# Singular Value Decomposition Structure

(Redirected from SVD Structure)

A Singular Value Decomposition Structure is a matrix decomposition $\displaystyle{ UΣV^T }$ for $\displaystyle{ m \times n }$ matrix $\displaystyle{ M }$, where $\displaystyle{ U }$ is a $\displaystyle{ m \times k }$ orthonormal matrix (with left singular vectors), $\displaystyle{ \Sigma }$ being a $\displaystyle{ k \times k }$ nonnegative diagonal matrix (of singular values), and $\displaystyle{ V^T }$ being an $\displaystyle{ n \times k }$ orthonormal matrix (with right singular vectors).

• Context:
• Example(s):
• $\displaystyle{ \begin{bmatrix}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}}\\ \frac{2}{\sqrt{5}} & \frac{-1}{\sqrt{5}}\end{bmatrix} \begin{bmatrix}\sqrt{125} & 0\\0 & 0\end{bmatrix} \begin{bmatrix}0.8 & 0.6\\0.6 & -0.8\end{bmatrix} }$, for $\displaystyle{ \operatorname{SVD}\left(\begin{bmatrix}4 & 3\\8 & 6\end{bmatrix}\right) }$.

• $\displaystyle{ \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} 4 & 0 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & \sqrt{5} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ \sqrt{0.2} & 0 & 0 & 0 & \sqrt{0.8} \\ 0 & 0 & 0 & 1 & 0 \\ -\sqrt{0.8} & 0 & 0 & 0 & \sqrt{0.2} \end{bmatrix} \ , \text{for} \ \operatorname{SVD}\left(\begin{bmatrix} 1 & 0 & 0 & 0 & 2 \\ 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 & 0 \end{bmatrix} \right) }$
• Counter-Example(s):
• See: Singular Value Decomposition Task.

## References

### 2012

• (Golub & Van Loan, 2012) ⇒ Gene H. Golub, and Charles F. Van Loan. (2012). “Matrix Computations (4th Ed.)." Johns Hopkins University Press. ISBN:1421408597
• QUOTE: If $\displaystyle{ A }$ is a real m-by-n matrix, then there exist orthogonal matrices :$\displaystyle{ U = \bigl [ u_1,...,u_m \bigr] \in \mathbb{R}^{m \times m} \ \text{ and } \ V = \bigl [ v_1,...,v_n \bigr] \in \mathbb{R}^{n \times n} }$ such that :$\displaystyle{ U^TAV = \Sigma = \operatorname{diag}(\sigma_1,...,\sigma_p) \in \mathbb{R}^{m \times n} \ = \ \operatorname{min}\{m,n\} }$ where $\displaystyle{ σ_1 ≥ σ_2 ≥ … ≥ σ_p ≥ 0 }$. ...

... The $\displaystyle{ σ_i }$ are the singular values of $\displaystyle{ A }$, the $\displaystyle{ u_i }$ are the left singular vectors of $\displaystyle{ A }$, and the $\displaystyle{ v_i }$ are right singular vectors of $\displaystyle{ A }$. Separate visualizations of the SVD are required depending upon whether $\displaystyle{ A }$ has more rows or columns. Here are the 3-by-2 and 2-by-3 examples: :$\displaystyle{ \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ u_{21} & u_{22} & u_{23} \\ u_{31} & u_{32} & u_{33} \end{bmatrix}^T \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{bmatrix} \begin{bmatrix} v_{11} & v_{12} \\ v_{21} & v_{22} \end{bmatrix} = \begin{bmatrix} \sigma_{1} & 0 \\ 0 & \sigma_{2} \\ 0 & 0 \end{bmatrix}. }$ :$\displaystyle{ \begin{bmatrix} u_{11} & u_{12} \\ u_{21} & u_{22} \end{bmatrix}^T \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \begin{bmatrix} v_{11} & v_{12} & v_{13} \\ v_{21} & v_{22} & v_{23} \\ v_{31} & v_{32} & v_{33} \end{bmatrix} = \begin{bmatrix} \sigma_{1} & 0 & 0 \\ 0 & \sigma_{2} & 0 \end{bmatrix} . }$