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The singular values of a matrix A can be viewed as describing the geometry of AB, where AB is the image of the euclidean ball under the linear transformation A. In particular, AB is an elipsoid, and the singular values of A describe the length of its major axes.

More generally, what do the singular values of a matrix say about the geometry of the image of other objects? How about the unit L1 ball? This will be some polytope: is there some natural way to describe this shape in terms of singular values, or other properties of matrix A?

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It's all about how the object of interest looks after you choose the orthogonal base corresponding to the singular decomposition of A. Then, is only a matter of stretching, just as with the euclidean ball. In this special case it's so simple, because the ball looks the same in all orthogonal bases. But since orthogonal transformation is only about rotation and reflection, the singular values then again describe the stretching of you object after the appropriate transformation.

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