Eigenvalues and Eigenvectors: Practice Questions
1. Find the eigenvalues and eigenvectors of the matrix:
A = [[2, 1, 0], [1, 2, 1], [0, 1, 2]]
2. Find the eigenvalues of the matrix:
A = [[4, 1, -2], [1, 3, 0], [-2, 0, 3]]
3. Determine whether the matrix is diagonalizable. If yes, find a diagonal matrix D:
A = [[1, 2, 2], [2, 1, 2], [2, 2, 1]]
4. Find the characteristic polynomial of the matrix:
A = [[0, 1, 0], [0, 0, 1], [-6, 11, -6]]
5. Compute the eigenvalues and eigenvectors of the matrix:
A = [[3, 0, 0], [0, 2, 1], [0, 1, 2]]
6. Find the eigenvalues and eigenvectors of the symmetric matrix:
A = [[1, 2, 3], [2, 1, 2], [3, 2, 1]]
7. Determine the eigenvalues of the matrix:
A = [[5, 4, 2], [0, 1, 0], [0, 0, 3]]
8. Find the eigenvectors for the given eigenvalue lambda = 2:
A = [[2, 0, 0], [1, 2, 0], [0, 0, 3]]
9. Show that the matrix has a repeated eigenvalue:
A = [[6, 2, 1], [2, 3, 1], [1, 1, 1]]
10. Find the eigenvalues and eigenvectors of the matrix:
A = [[0, -1, 0], [1, 0, 0], [0, 0, 1]]