Discussion
Let \(A = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 1 & 2 \end{pmatrix}\). Then, singular values of A :
Answer: D
The singular values are √7 and √3.
The singular values of A are the square roots of the non-zero eigenvalues of \(AA^{T}\):
\[ AA^{T} = \begin{pmatrix} 5 & 2 \\ 2 & 5 \end{pmatrix} \]
Its eigenvalues satisfy \((5 - \lambda)^{2} = 4\), so \(\lambda = 7\) or \(\lambda = 3\), giving singular values \(\sqrt{7}\) and \(\sqrt{3}\). (\(A^{T}A\) is 3 × 3 with eigenvalues 7, 3 and 0, and the 0 contributes no singular value for a rank-2 matrix.)
'7 and 3' forgets the square root. Singular values exist for every matrix, rectangular or not, and '√3, 2 and 1' does not come from \(AA^{T}\).
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