Eigenvalues and Eigenvectors
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. ...