Question

Find the SVD of the following matrices:

i). \begin{bmatrix} -1 & 1 & 1\\ 1 & 1 & 0\\ \end{bmatrix}

ii). \begin{bmatrix} -1 & 1 \\ 1 & 1 \\ 1 & 2 \end{bmatrix}

06 Feb 2024
Answer :
Word Count : 1003

To find the Singular Value Decomposition (SVD) of a matrix, we need to compute its singular values and corresponding left and right singular vectors.

i). For the matrix \(\begin{bmatrix} -1 & 1 & 1\\ 1 & 1 & 0\\ \end{bmatrix}\):

1. Compute \(A^TA\):
\[
A^TA = \begin{bmatrix} -1 & 1 & 1\\ 1 & 1 & 0\\ \end{bmatrix}^T \begin{bmatrix} -1 & 1 & 1\\ 1 & 1 & 0\\ \end{bmatrix} = \begin{bmatrix} 2 & 0 & -1\\ 0 & 2 & 1\\ -1 & 1 & 1 \end{bmatrix}
\]

2. Compute the eigenvalues and eigenvectors of \(A^TA\).

   The eigenvalues of \(A^TA\) are \(\lambda_1 = 3\), \(\lambda_2 = 2\), and \(\lambda_3 = 0\).

   For \(\lambda_1 = 3\):
   \[
   A^TA - 3I = \begin{bmatrix} -1 & 0 & -1\\ 0 & -1 & 1\\ -1 & 1 & -2 \end{bmatrix} \implies v_1 = \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}
   \]

   For \(\lambda_2 = 2\):
   \[
   A^TA - 2I = \begin{bmatrix} 0 & 0 & -1\\ 0 & 0 & 1\\ -1 & 1 & -1 \end{bmatrix} \implies v_2 = \begin{bmatrix} 1\\ -1\\ 0 \end{bmatrix}
   \]

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