User:Scut723/sandbox/Singular Value Decomposition

Singular Value Decomposition

The singular value decomposition (SVD) is one of the most powerful tools in theoretical and numerical linear algebra. The utility comes from three basic properties:

Every matrix has an SVD. The SVD provides an orthonormal resolution for the four invariant subspaces. The SVD provides an ordered list of singular values.

Existence
Every matrix has a singular value decomposition. Given a matrix $$\mathbf{A} \in \mathbb{C}^{m \times n}_{\rho}$$, that is, with $$m$$ rows, $$n$$ columns, and rank $$\rho$$, the SVD can be written as
 * $$\mathbf{A} = \mathbf{U} \Sigma \mathbf{V}^{*}$$,

where
 * $$\mathbf{U} \in \mathbb{C}^{m \times m}$$ resolves the column space,
 * $$\mathbf{V} \in \mathbb{C}^{n \times n}$$ resolves the row space,
 * $$\mathbf{\Sigma} \in \mathbb{R}^{m \times n}_{\rho}$$ contains the singular values.

The domain matrices are unitary:


 * $$\mathbf{U}\mathbf{U}^{*} = \mathbf{U}^{*}\mathbf{U} = \mathbf{I}_{m}$$
 * $$\mathbf{V}\mathbf{V}^{*} = \mathbf{V}^{*}\mathbf{V} = \mathbf{I}_{n}$$

Uniqueness
The singular values are unique, therefore the matrices $$\mathbf{S}$$ and $$\Sigma$$ are unique. Typically the domain matrices are not unique. For example, there could be two different decompositions such that


 * $$\mathbf{A} = \mathbf{U}_{1} \mathbf{\Sigma} \mathbf{V}_{1}^{*} = \mathbf{U}_{2} \mathbf{\Sigma} \mathbf{V}_{2}^{*}$$