Singular value decomposition

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any ${\displaystyle m\times n}$ matrix via an extension of polar decomposition. It has many useful applications in signal processing and statistics.

Formally, the singular value decomposition of an ${\displaystyle m\times n}$ real or complex matrix ${\displaystyle \mathbf {M} }$ is a factorization of the form ${\displaystyle \mathbf {U\Sigma V^{*}} }$, where ${\displaystyle \mathbf {U} }$ is an ${\displaystyle m\times m}$ real or complex unitary matrix, ${\displaystyle \mathbf {\Sigma } }$ is a ${\displaystyle m\times n}$ rectangular diagonal matrix with non-negative real numbers on the diagonal, and ${\displaystyle \mathbf {V} }$ is an ${\displaystyle n\times n}$ real or complex unitary matrix. The diagonal entries ${\displaystyle \sigma _{i}}$ of ${\displaystyle \mathbf {\Sigma } }$ are known as the singular values of ${\displaystyle \mathbf {M} }$. The columns of ${\displaystyle \mathbf {U} }$ and the columns of ${\displaystyle \mathbf {V} }$ are called the left-singular vectors and right-singular vectors of ${\displaystyle \mathbf {M} }$, respectively.

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