To solve for the eigenvectors of the given matrix, we'll follow these steps:

Steps to Solve for Eigenvectors:

1. Find the Eigenvalues:

   • Solve the characteristic equation $\text{det}(A - \lambda I) = 0$ where $A$ is the given matrix, $\lambda$ is the eigenvalue, and $I$ is the identity matrix.

2. Find the Eigenvectors:

   • For each eigenvalue $\lambda$, solve the equation $(A - \lambda I)\mathbf{v} = 0$ to find the corresponding eigenvectors $\mathbf{v}$.

Let's proceed with these steps for the matrix:

0 & -2 & 4 \\ -2 & 1 & 8 \\ -2 & -1 & 4 \end{pmatrix}$$The eigenvalues and their corresponding eigenvectors for the given matrix are: ### Eigenvalues: 1. $$\lambda_1 = 0.9265 + 3.0512i$$ 2. $$\lambda_2 = 0.9265 - 3.0512i$$ 3. $$\lambda_3 = 3.14696$$ ### Corresponding Eigenvectors: 1. For $$\lambda_1 = 0.9265 + 3.0512i$$: $$\mathbf{v}_1 = \begin{pmatrix} -0.3641 - 0.4624i \\ -0.7059 \\ -0.0845 - 0.3849i \end{pmatrix}$$ 2. For $$\lambda_2 = 0.9265 - 3.0512i$$: $$\mathbf{v}_2 = \begin{pmatrix} -0.3641 + 0.4624i \\ -0.7059 \\ -0.0845 + 0.3849i \end{pmatrix}$$ 3. For $$\lambda_3 = 3.14696$$: $$\mathbf{v}_3 = \begin{pmatrix} -0.3920 \\ 0.9083 \\ 0.1458 \end{pmatrix}$$ ### Interpretation: - The matrix has complex eigenvalues and eigenvectors, which indicates that the matrix may represent a transformation involving rotation and scaling in a complex plane. - The third eigenvalue is real, and its corresponding eigenvector is purely real, which suggests a different kind of transformation (likely a scaling without rotation). Would you like further details on how these results were derived or any other questions? --- **Related Questions:** 1. How do complex eigenvalues relate to rotations and scaling in transformations? 2. What is the significance of having both real and complex eigenvalues in a matrix? 3. How do you determine the stability of a system using eigenvalues? 4. Can eigenvectors be used to diagonalize a matrix? 5. What is the Jordan canonical form of a matrix with complex eigenvalues? 6. How are eigenvalues and eigenvectors applied in quantum mechanics? 7. What are the practical applications of complex eigenvalues in engineering? 8. How do you interpret the geometric significance of complex eigenvectors? **Tip:** When eigenvalues are complex, the corresponding eigenvectors can have complex components, representing more intricate transformations such as rotations in complex space.