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Singular Value Decomposition


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For any real $ m\times n$ matrix $ A$ there exist unitary matrices $ U$ and $ V$ 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 $ A^* A$, $ k$ is the rank of $ A$, and the columns of $ U$ and $ V$ are eigenvectors of $ AA^*$ and $ A^* A$, respectively. Moreover,

$\displaystyle Av_j = \sigma_j u_j
$

for $ 1\leq j \leq k$.

In the special case of a real matrix $ A$, the matrices $ U$ and $ V$ are orthogonal.

see also:


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  automatically generated 6/29/2007