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The singular value decomposition is a factorization of a matrix

into

. A vector

is first rotated by an angle β via

, then

is scaled by a diagonal matrix

to form

. Finally the vector

is rotated by an angle ω to form

Note that U and V are orthogonal matrices and thus do not alter the length of the vectors. In 2D the diagonal matrix Σ has the form

here

are the singular values of the matrix.

Contributed by: Chris Maes

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Singular Value Decomposition