Word Count : 531

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To find the Singular Value Decomposition (SVD) of the given matrices, we follow these steps: ### (i) Matrix \( A = \begin{bmatrix} -1 & 1 & 1 \\ 1 & 1 & 0 \end{bmatrix} \) 1. Compute \( A^T A \): \[ A^T A = \begin{bmatrix} -1 & 1 \\ 1 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} -1 & 1 & 1 \\ 1 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 2 & 0 & -1 \\ 0 & 2 & 1 \\ -1 & 1 & 1 \end{bmatrix} \] 2. Find the eigenvalues and eigenvectors of \( A^T A \): - The eigenvalues are \( \lambda_1 = 3 \), \( \lambda_2 = 2 \), and \( \lambda_3 = 0 \). - The corresponding eigenvectors are: \[ v_1 = \begin{bmatrix} ___ ________ ____ _________ _____ ____ _____ ________ ____ ____.
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