Eigenvalues

Eigenvalues and Eigenvectors of Real Matrices Exercise 1 : Eigenvalues — 2x2 real matrix For $A = \begin{pmatrix} 4 & 2 \ 1 & 3 \end{pmatrix}$ find the eigenvalues. Exercise 2 : Real vs Complex Eigenvalues — 2x2 question Is it possible for a real $2 \times 2$ matrix to have one real and one complex eigenvalue? Exercise 3 : No Real Eigenvalues — rotation-like matrix Prove that there are no real eigenvalues of the matrix $A = \begin{pmatrix} 3 & 2 \ -2 & 1 \end{pmatrix}$ ...

Eigenvalues and Eigenvectors Exercise 1 Confirm by multiplication that x is an eigenvector of A, and find the corresponding eigenvalue. \( A = \begin{bmatrix} 1 & 2 \\ 3 & 2 \end{bmatrix}, \quad x = \begin{bmatrix} 1 \\ -1 \end{bmatrix} \) \( A = \begin{bmatrix} 5 & -1 \\ 3 & 2 \end{bmatrix}, \quad x = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \) \( A = \begin{bmatrix} 4 & 0 & 0 \\ 2 & 3 & 2 \\ 1 & 0 & 4 \end{bmatrix}, \quad x = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \) \( A = \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix}, \quad x = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \) Exercise 2 Find the characteristic equation, the eigenvalues, and bases for the eigenspaces of the matrix. ...