The singular value decomposition (SVD) enables you to express any linear transformation as a rotation, followed by a scaling, followed by another rotation. The SVD states that any n by p matrix X can be written as follows:
Diag(Λ) is an r by r diagonal matrix with positive diagonal elements given by the column vector where .
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Note: There are various conventions in the literature regarding the dimensions of the matrices U, V, and the matrix containing the singular values. However, the differences have no practical impact on the decomposition up to the rank of X.
For further details, see Press et al. (1998, Section 2.6).