Questions tagged [svd]

In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix.

Formally, the singular value decomposition of an $m \times n$ real or complex matrix $M$ is a factorisation of the form $UAV^*$ where $U$ is an $m\times m$ real or complex unitary matrix, $A$ is an $m\times n$ rectangular diagonal matrix with non-negative real numbers on the diagonal, and $V$ is an $n\times n$ real or complex unitary matrix.

The singular-value decomposition can be computed using the following observations:

• The left-singular vectors of $M$ are a set of orthonormal eigenvectors of $MM^*$.
• The right-singular vectors of $M$ are a set of orthonormal eigenvectors of $M^*M$.
• The non-zero singular values of $M$ (found on the diagonal entries of $A$) are the square roots of the non-zero eigenvalues of both $M^*M$ and $MM^*$.

Source: Wikipedia.