Ellipsoids (7): relationship with singular value decomposition (SVD)

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 09:41

Summary:

This video discusses the relationship between the singular value decomposition $L=US{V}^T$ of a matrix $L$ with the geometry of the linear transformation with said matrix $L$ of a unit sphere $x=L\nu$, $\parallel \nu {\parallel }_2\le 1$.

It is shown that the directions of the semiaxes are the columns of $U$, that the lengths of the semiaxes are the singular values (diagonal of $S$), and that the columns of $V$ (input directions) are the points of the unit sphere that multiplied by $L$ become the endpoints of the semiaxes of the ellipsoid.

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