Past exam of the numerical linear algebra course of the University of Oxford / 2011 / 1 / a / Solution

A real singular value decomposition has the form $A=U\Sigma V^T$, where $U\in {\mathbb{R}}^{m\times m}$ and $V\in {\mathbb{R}}^{n\times n}$ are orthogonal matrices, and $\Sigma$ is a rectangular diagonal matrix with entries ${\sigma }_1\ge \cdots \ge {\sigma }_{\min (m,n)}\ge 0$. These entries are the singular values. Their squares are the eigenvalues of $A^{T}A$, omitting the additional zero eigenvalues when $n>m$. The positive singular values number exactly the matrix rank.