Questions: Find the eigenvalues of A, and find a basis for each eigenspace. A=[ 1 -0.5 1.6 0.2 ]

Solution Steps

To find the eigenvalues of matrix \( A \), we need to solve the characteristic equation \( \text{det}(A - \lambda I) = 0 \). Once we have the eigenvalues, we can find the corresponding eigenvectors by solving the equation \( (A - \lambda I)v = 0 \) for each eigenvalue \( \lambda \).

The eigenvalues of the matrix \( A = \begin{bmatrix} 1 & -0.5 \\ 1.6 & 0.2 \end{bmatrix} \) are calculated from the characteristic polynomial. The eigenvalues are given by: \[ \lambda_1 = 0.6 + 0.8i, \quad \lambda_2 = 0.6 - 0.8i \]

The corresponding eigenvectors for each eigenvalue can be determined from the equation \( (A - \lambda I)v = 0 \).

For \( \lambda_1 = 0.6 + 0.8i \): The eigenvector is: \[ v_1 = \begin{bmatrix} 0.2182 + 0.4364i \\ 0.8729 \end{bmatrix} \]

For \( \lambda_2 = 0.6 - 0.8i \): The eigenvector is: \[ v_2 = \begin{bmatrix} 0.2182 - 0.4364i \\ 0.8729 \end{bmatrix} \]

Final Answer