Mathematics-Online course: Linear Algebra-Normal Forms-Singular Value Decomposition-Singular Value Decomposition

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	Mathematics-Online course: Linear Algebra - Normal Forms - Singular Value Decomposition
	Singular Value Decomposition

For any real $m\times n$ matrix

there exist unitary matrices

and

with

$\displaystyle U^* A V = S = \left(\begin{array}{ccc} s_1 & & 0 \\ & s_2 & \\ 0 & & \ddots \end{array}\right).$

The singular values

$\displaystyle s_1\ge s_2\ge\cdots\ge s_k>s_{k+1}=\cdots=0$

are the square roots of the eigenvalues of

is the rank of

, and the columns of

and

are eigenvectors of

and

, respectively. Moreover,

$\displaystyle Av_j = \sigma_j u_j$

for $1\leq j \leq k$ .

In the special case of a real matrix , the matrices and are orthogonal.

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automatically generated 4/21/2005