Assignment-2 on Matrix

A. Eigenvalues and Eigenvectors

1. Find the eigenvalues and eigenvectors of the matrix \( A = \begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix} \).
2. Find the eigenvalues and eigenvectors of the matrix \( B = \begin{bmatrix}1 & 2 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4\end{bmatrix} \).
3. Find the eigenvectors corresponding to the eigenvalue 3 of \( C = \begin{bmatrix}1 & 1 \\ 0 & 3\end{bmatrix} \).
4. Determine whether the matrix \( D = \begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix} \) is diagonalizable and justify.
5. Diagonalize the matrix \( E = \begin{bmatrix}5 & 4 \\ 2 & 3\end{bmatrix} \) and verify its eigenvalues.
6. Diagonalize the matrix \( L = \begin{bmatrix}2 & 0 & 0 \\ 0 & 3 & 4 \\ 0 & 0 & 5\end{bmatrix} \).
7. Diagonalize the matrix \( M = \begin{bmatrix}1 & 2 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 3\end{bmatrix} \).
8. Find the eigenvalues and eigenvectors of the matrix \( A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{bmatrix} \).
9. Find the eigenvalues and eigenvectors of the matrix \( B = \begin{bmatrix}2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2\end{bmatrix} \).
10. Find the eigenvalues and eigenvectors of the matrix \( C = \begin{bmatrix}4 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 4\end{bmatrix} \).
11. Find the eigenvalues and eigenvectors of the matrix \( D = \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 6 & -11 & 6\end{bmatrix} \).
12. Determine whether the matrix \( E = \begin{bmatrix}3 & 1 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 3\end{bmatrix} \) is diagonalizable and find its eigenvectors.

B. Cayley-Hamilton Theorem

1. Verify the Cayley-Hamilton theorem for the matrix \( G = \begin{bmatrix}1 & 1 \\ 4 & 1\end{bmatrix} \).
2. Verify the Cayley-Hamilton theorem for the matrix \( H = \begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix} \).
3. Verify the Cayley-Hamilton theorem for the matrix \( I = \begin{bmatrix}2 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2\end{bmatrix} \).
4. Use the Cayley-Hamilton theorem to find the inverse of \( J = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix} \), if it exists.

C. Quadratic Forms, Nature, and Canonical Form

1. Find the canonical form of the quadratic form \( Q_1(x,y) = 3x^2 + 4xy + y^2 \).
2. Find the nature of \( Q_2(x,y) = x^2 + 2xy + y^2 \) (positive definite, negative definite, or indefinite).
3. Find the canonical form of \( Q_3(x,y) = 4x^2 - 4xy + y^2 \).
4. Determine the nature of \( Q_4(x,y,z) = x^2 + y^2 - z^2 \).
5. Determine whether \( Q_5(x,y) = x^2 - y^2 \) is positive definite, negative definite, or indefinite.
6. Find the canonical form of \( Q_1(x,y,z) = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz \).
7. Find the nature of \( Q_2(x,y,z) = x^2 + 2y^2 + 3z^2 + 2xy - 2xz + 4yz \) (positive definite, negative definite, or indefinite).
8. Find the canonical form of \( Q_3(x,y,z) = 2x^2 - y^2 + z^2 + 2xy - 2xz + 2yz \).
9. Determine the nature of \( Q_4(x,y,z) = x^2 + y^2 - z^2 + 2xy - 2yz \).