Suppose matrix $\mathbf{A}\in\mathbb{C}^{m\times n}$ has eigenvalues $\lambda_i$, where $i=1,\ldots,\min(m,n)$. Then, the singular values of $\mathbf{A}$ are the nonnegative numbers:

$\sigma_i(\mathbf{A}) \triangleq \lambda_i^{1/2}(\mathbf{AA}^\ast) = \lambda_i^{1/2}(\mathbf{A}^\ast\mathbf{A}).$

A singular value decomposition (SVD) of a matrix $\mathbf{A}\in\mathbb{C}^{m\times n}$ is the factorisation:

$\mathbf{A} = \mathbf{U\Sigma V}^\ast, \quad \mathbf{\Sigma} = \text{diag}(\sigma_1,\ldots,\sigma_p) \in \mathbb{R}^{m\times n}, \quad p=\min\{m,n\},$

where $\sigma_1\geq\sigma_2\geq\cdots\geq\sigma_p\geq0$, $\mathbf{U}$ and $\mathbf{V}$ are unitary.

The diagonal entries of $\mathbf{\Sigma}$, namely $\sigma_1,\ldots,\sigma_p$, are the singular values of $\mathbf{A}$.

The columns of $\mathbf{U}$ are the left singular vectors of $\mathbf{A}$.

The columns of $\mathbf{V}$ are the right singular vectors of $\mathbf{A}$.