Definition of singular value decomposition (SVD)

Created over 8 years ago, updated 10 days ago

Definition: The singular value decomposition (SVD) of an $m \times n$ matrix $A$ is a factorization:

$$A = U \Sigma V^*$$

where:

$U$ is an $m \times m$ unitary (orthogonal if real) matrix
$V$ is an $n \times n$ unitary (orthogonal if real) matrix
$\Sigma$ is an $m \times n$ diagonal matrix with non-negative entries $\sigma_1 \geq \sigma_2 \geq \cdots \geq 0$ called singular values

Key facts:

Singular values are the square roots of eigenvalues of $A^*A$ (or $AA^*$)
Columns of $U$ are left singular vectors, columns of $V$ are right singular vectors
The number of nonzero singular values equals the rank of $A$
The SVD exists for every matrix (real or complex, square or rectangular)

Applications: data compression, PCA, pseudoinverse computation, least squares, image processing.