= 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 $\sigma_1\geq\cdots\geq\sigma_{\min(m,n)}\geq0$. These entries are the <singular values>. Their squares are the <eigenvalues> of $A^TA$, omitting the additional zero <eigenvalues> when $n>m$. The positive <singular values> number exactly the <matrix rank>.
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