singular Value Decomposition(SVD)

From statwiki
Jump to navigation Jump to search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

For a more in depth reference see: SVD

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

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


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

[math]\displaystyle{ \ S^T = S }[/math]


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

[math]\displaystyle{ \ U^TU=I }[/math]

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

[math]\displaystyle{ \ V^TV=I }[/math]

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


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

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


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

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

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

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


In MATLAB execute:

 [USV] = svd(X)