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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.

The singular value decomposition is the most powerful - and most expensive - decomposition tool in linear algebra. The power comes from the resolution of the four fundamental subspaces as well as the eigenvalues.

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

${\displaystyle \mathbf {A} =\mathbf {U} \Sigma \mathbf {V} ^{*}}$,

where

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

The domain matrices are unitary:

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

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

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

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

The [Fundamental Theorem of Linear Algebra] states that a matrix ${\displaystyle \mathbf {A} \in \mathbb {C} _{\rho }^{m\times n}}$ induces a row space (or domain) ${\displaystyle \mathbb {C} ^{n}}$ and a column space (or codomain) ${\displaystyle \mathbb {C} ^{m}}$. The row space and the column space each have an orthogonal decomposition into a range space and a null space:

• ${\displaystyle \mathbb {C} ^{n}}$ = ${\displaystyle {\color {Blue}{\overline {{\mathcal {R}}\left(\mathbf {A} ^{*}\right)}}}}$ ${\displaystyle \oplus }$ ${\displaystyle {\color {Red}{{\mathcal {N}}\left(\mathbf {A} \right)}}}$ (domain),

• ${\displaystyle \mathbb {C} ^{m}}$ = ${\displaystyle {\color {Blue}{\overline {{\mathcal {R}}\left(\mathbf {A} \right)}}}}$ ${\displaystyle \oplus }$ ${\displaystyle {\color {Red}{{\mathcal {N}}\left(\mathbf {A} ^{*}\right)}}}$ (codomain),

where the overbear represents the set closure required in infinite dimensional spaces.

${\displaystyle {\color {Blue}{x^{2}}}+{\color {Orange}{2x}}-{\color {LimeGreen}{1}}}$

${\displaystyle x^{2}+y^{2}+z^{2}=1}$

1

${\displaystyle {\begin{pmatrix}x&y\\z&v\end{pmatrix}}}$

Casting the SVD in block structure emphasizes its subspace decomposition;

${\displaystyle \mathbf {A} =\mathbf {U} \Sigma \mathbf {V} ^{*}=\left[{\begin{array}{cc}{\color {Blue}{\mathbf {U} _{\mathcal {R}}}}&{\color {Red}{\mathbf {U} _{\mathcal {N}}}}\\\end{array}}\right]\left[{\begin{array}{c|c}\mathbf {S} &\mathbf {0} \\\hline \mathbf {0} &\mathbf {0} \end{array}}\right]\left[{\begin{array}{c}{\color {Blue}{\mathbf {V} _{\mathcal {R}}^{*}}}\\{\color {Red}{\mathbf {V} _{\mathcal {N}}^{*}}}\end{array}}\right]}$

The mapping action of a matrix demonstrates the geometry of the SVD. A matrix is an operator which maps an ${\displaystyle n-}$vector into an ${\displaystyle m-}$vector

${\displaystyle \mathbf {A} \colon \mathbb {C} ^{n}\mapsto \mathbb {C} ^{m}}$

${\displaystyle m=n=\rho =2}$

${\displaystyle \left[{\begin{array}{cr}1&-1\\2&2\\\end{array}}\right]}$