singular Value Decomposition(SVD)

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For a more in depth reference see: SVD

Any matrix [math]\ X [/math] can be decomposed into three matrices:

[math]\ {X}_{d \times n} = {U}_{d \times d}{S}_{d \times d}{V^T}_{d \times n} [/math]


Where [math]\ S [/math] is a diagonal matrix and thus, has the following property:

[math]\ S^T = S [/math]


And [math]\ U [/math] and [math]\ V [/math] are both orthonormal matrices and thus, have the following properties:

[math]\ U^TU=I [/math]

[math]\ U^T=U^{-1} [/math]

[math]\ V^TV=I [/math]

[math]\ V^T=V^{-1} [/math]


The [math]\ S, U [/math] and [math]\ V [/math] matrices are constructed in the following manner:

[math]\ S = [/math] eigenvalues of [math]\ X^TX [/math] = eigenvalues of [math]\ XX^T[/math]


[math]\ U[/math] represents the left singular vectors of [math]\ X[/math]

[math]\ V[/math] represents the right singular vectors of [math]\ X[/math]

[math]\ U = [/math] eigenvectors of [math]\ X X^T[/math]

[math]\ V = [/math] eigenvectors of [math]\ X^T X[/math]


In MATLAB execute:

 [USV] = svd(X)