Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2007 / A / 1 / b / Solution

A full real singular value decomposition iswhere $\Sigma \in {\mathbb{R}}^{m\times n}$ has only diagonal entries, with ${\sigma }_1\ge \cdots \ge {\sigma }_{\min (m,n)}\ge 0$. These are the singular values. The number of positive singular values is the matrix rank. The columns of $V$ are right singular vectors and those of $U$ are left singular vectors, giving orthonormal input and output coordinate systems.