09. Singular Value Decomposition

31 - SVD

SVD

Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$ then
Every matrix $A\in {\mathbb{F}}^{m\times n}$ with $m\ge n$ can be written as

$$A=U\Sigma V^{\ast }$$

where

1. $U\in {\mathbb{F}}^{m\times n}$ is a matrix with orthonormal columns so $U^{\ast }U={\mathrm{Id}}_n$

2. $V\in {\mathbb{F}}^{n\times n}$ is a matrix with orthonormal columns so $V^{\ast }V={\mathrm{Id}}_n=V{V}^{\ast }$

3. $\Sigma$ is a diagonal matrix with non-negative diagonal entries ${\sigma }_1\ge {\sigma }_2\ge \cdots \ge {\sigma }_n\ge 0$

Proof

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Polar Decomposition corollary

Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$ then
Every matrix $A\in {\mathbb{F}}^{m\times n}$ with $m\ge n$ can be written as

$$A=U_{p}H$$

where

1. $U_p\in {\mathbb{F}}^{m\times n}$ has orthonormal columns
2. $H\in {\mathbb{F}}^{n\times n}$ is self-adjoint ($H^{\ast }=H$) and positive semi-definite

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