SVD: a example

Consider a matrix $A in mathbf{R}^{4 times 4}$ , with SVD

$A = U tilde{ {S}} V^T$

where tilde{ {S}} = mbox{bf diag}(10, 7, 0.1, 0.05) .

From the SVD, we can understand the behavior of the mapping x rightarrow Ax :

input components along directions and are amplified (by about a factor ) and come out mostly along plane spanned by , .
Input components along directions and are attenuated (by about a factor ).
The matrix is nonsingular.
For some applications you might say A is effectively rank .