singular Value Decomposition(SVD): Difference between revisions
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Any matrix <math>\ X </math> can be decomposed into three matrices: | 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> | <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: | Where <math>\ S </math> is a diagonal matrix and thus, has the following property: | ||
<math>\ S^T = S </math> | <math>\ S^T = S </math> | ||
And <math>\ U </math> and <math>\ V </math> are both orthonormal matrices and thus, have the following properties: | 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^TU=I </math> | ||
<math>\ U^T=U^{-1} </math> | <math>\ U^T=U^{-1} </math> | ||
<math>\ V^TV=I </math> | <math>\ V^TV=I </math> | ||
<math>\ V^T=V^{-1} </math> | <math>\ V^T=V^{-1} </math> | ||
The <math>\ S, U </math> and <math>\ V </math> matrices are constructed in the following manner: | 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>\ 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>\ U</math> represents the left singular vectors of <math>\ X</math> | ||
<math>\ V</math> represents the right 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>\ U = </math> eigenvectors of <math>\ X X^T</math> | ||
<math>\ V = </math> eigenvectors of <math>\ X^T X</math> | <math>\ V = </math> eigenvectors of <math>\ X^T X</math> | ||
Revision as of 10:34, 25 September 2011
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)