SVD: a 4×4 example

L. El Ghaoui

SVD: a 4×4 example

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

$\begin{gathered} A=U \tilde{S} V^T \\ \text { where } \tilde{S}=\operatorname{diag}(10,7,0.1,0.05) \text {. } \end{gathered}$

From the SVD, we can understand the behavior of the mapping $x \rightarrow A x$ :
– input components along directions $v_1$ and $v_2$ are amplified (by about a factor 10) and come out mostly along plane spanned by $u_1, u_2$ .
– Input components along directions $v_3$ and $v_4$ are attenuated (by about a factor 10 ).
– The matrix $A$ is nonsingular.
– For some applications you might say A is effectively rank 2.

SVD: a 4×4 example

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