Word Count : 570

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i) Consider the matrix ( A = \begin{bmatrix}-1 & 1 & 1 \ 1 & 1 & 0\end{bmatrix} ). To find its SVD, we compute ( A A^T ) and ( A^T A ). [ A A^T = \begin{bmatrix}-1 & 1 & 1 \ 1 & 1 & 0\end{bmatrix} \begin{bmatrix}-1 & 1 \ 1 & 1 \ 1 & 0\end{bmatrix} = \begin{bmatrix}(-1)^2+1^2+1^2 & (-1)(1)+1(1)+1(0) \ 1(-1)+1(1)+0(1) & 1^2+1^2+0^2 \end{bmatrix} = \begin{bmatrix}3 & 0 \ 0 & 2\end{bmatrix} ] The eigenvalues of ( A A^T ) are ( \lambda_1 = 3 ) and ( \lambda_2 = 2 ), so the singular values are ( \sigma_1 = \sqrt{3} ), ( \sigma_2 = \sqrt{2} ). The eigenvectors ____ ____ _______ ________ ________ __________ __________ ______ _______.
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