Question-48

Find the singular values of the following matrix:

\[\begin{equation*} \begin{bmatrix} -4 & -6\\ 3 & -8 \end{bmatrix} \end{equation*}\]

• \(\sigma_1 = 1, \sigma_2 = 0\)
• \(\sigma_1 = 10, \sigma_2 = 5\)
• \(\sigma_1 = 100, \sigma_2 = 25\)
• \(\sigma_1 = \sigma_2 = 0\)

Answer

• \(\sigma_1 = 1, \sigma_2 = 0\)
• \(\sigma_1 = 10, \sigma_2 = 5\)
• \(\sigma_1 = 100, \sigma_2 = 25\)
• \(\sigma_1 = \sigma_2 = 0\)

Solution

Let \(\displaystyle A=\begin{bmatrix} -4 & -6\\ 3 & -8 \end{bmatrix}\). Then, \(\displaystyle A^{T} A=\begin{bmatrix} 25 & 0\\ 0 & 100 \end{bmatrix}\). The singular values of \(\displaystyle A\) are the positive square roots of the eigenvalues of \(\displaystyle A^{T} A\). Therefore, \(\displaystyle \sigma _{1} =10\) and \(\displaystyle \sigma _{2} =5\).