Linear Transformations Map Balls onto Ellipses: The Singular Value Proof
We prove that the image of the unit ball in n-dimensional Euclidean space under a linear transformation is an ellipsoid in n-dimensional Euclidean space. The lengths of the principal axes of the ellipsoid are twice the singular values of the linear operator. The Polar Decomposition provides us with the unitary mapping which will realign the ellipsoid with the standard axes.
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