Calculate the eigenvalues and eigenvectors of the matrix \(\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}\) and verify that they satisfy the eigenvalue equation.
Section 9.5 Exercises
Checkpoint 9.5.1.
Checkpoint 9.5.2.
Perform singular value decomposition on the matrix \(\begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \end{bmatrix}\) and reconstruct the matrix using the first two singular values.
Checkpoint 9.5.3.
Create a \(5 \times 5\) random matrix, add noise, and then use SVD for denoising. Compare the denoising effects of different numbers of singular values.
Checkpoint 9.5.4.
Explain why the SVD of a symmetric matrix is equivalent to its eigendecomposition. Provide mathematical proof and numerical examples.
