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Howtonotwin (talk | contribs) m →Based on variational characterization: typo: section is about vectors u,v not the matrices U,V |
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=== Based on variational characterization ===
{{anchor|vch}}The singular values can also be characterized as the maxima of {{tmath|\mathbf u^\mathrm{T} \mathbf M \mathbf v,}} considered as a function of {{tmath|\mathbf
Let {{tmath|\mathbf M}} denote an {{tmath|m \times n}} matrix with real entries. Let {{tmath|S^{k-1} }} be the unit <math>(k-1)</math>-sphere in <math> \mathbb{R}^k </math>, and define <math>\sigma(\mathbf{u}, \mathbf{v}) = \mathbf{u}^\operatorname{T} \mathbf{M} \mathbf{v},</math> <math>\mathbf{u} \in S^{m-1},</math> <math>\mathbf{v} \in S^{n-1}.</math>
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