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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 \( v_4 \).