SVD:  A 4×4 example

Consider a matrix $A$ in $\mathbb{R}^{4 \times 4}$ , with SVD given by

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

where

$\tilde{S} = \mathbf{diag}(10, 7, 0.1, 0.05).$

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

Input components along directions corresponding to $v_1$ and $v_2$ are amplified (by factors of 10 and 7, respectively) and come out mostly along the plane spanned by $u_1$ and $u_2$ .
Input components along directions corresponding to $v_3$ and $v_4$ are attenuated (by factors of 0.1 and 0.05, respectively).
The matrix $A$ is nonsingular.
For some applications, it might be appropriate to consider $A$ as effectively rank 2, given the significant attenuation for components along $v_3$ and $v_4$ .

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