User:Jarle Pahr/SVD: Difference between revisions

Revision as of 04:04, 13 February 2014

Notes on Singular Value Decomposition (SVD):

Eigenvectors and eigenvalues:

An eigenvector of a matrix is A is a non-zero vector [math]\displaystyle{ \overrightarrow v }[/math] that satisfies the equation

[math]\displaystyle{ A\overrightarrow v = \lambda \overrightarrow v }[/math]

Where [math]\displaystyle{ \lambda }[/math] is a scalar. [math]\displaystyle{ \lambda }[/math] is called an eigenvalue.

Singular values:

Any mxn matrix A can be factored as:

[math]\displaystyle{ A = U\sum {V^T} }[/math] where:

[math]\displaystyle{ U }[/math] is an mXm orthogonal matrix where the columns are the eigenvectors of [math]\displaystyle{ AA^T }[/math]
[math]\displaystyle{ V }[/math] is an nXn orthogonal matrix where the columns are the eigenvectors of [math]\displaystyle{ A^{T}A }[/math]
[math]\displaystyle{ \sum{} }[/math] is an mXn diagonal matrix where the [math]\displaystyle{ r }[/math] first diagona elements are the square roots of the eigenvalues of [math]\displaystyle{ A^{T}A }[/math], also called the singular values of A. Singular values are always real and positive.

The rank of a matrix A equals the number of singular values of A.

math.stackexchange.com/questions/261801/how-can-you-explain-the-singular-value-decomposition-to-non-specialists

@@ Line 2: / Line 2: @@
 See also http://en.wikipedia.org/wiki/Singular_value_decomposition
+Eigenvectors and eigenvalues:
+An eigenvector of a matrix is A is a non-zero vector <math>\overrightarrow v</math> that satisfies the equation
+<math>A\overrightarrow v  = \lambda \overrightarrow v</math>
+Where <math>\lambda</math> is a scalar. <math>\lambda</math> is called an eigenvalue.
 Singular values:
@@ Line 15: / Line 24: @@
 *<math>V</math> is an nXn orthogonal matrix where the columns are the eigenvectors of <math>A^{T}A</math>
 *<math>\sum{}</math> is an mXn diagonal matrix where the <math>r</math> first diagona elements are the square roots of the eigenvalues of <math>A^{T}A</math>, also called the singular values of A. Singular values are always real and positive.
+The rank of a matrix A equals the number of singular values of A.
 http://www.uwlax.edu/faculty/will/svd/