1. **Problem Statement:** We need to find the eigenvalues and eigenvectors of the stiffness matrix $$K = \begin{bmatrix} k_x & 0 \\ 0 & k_y \end{bmatrix}$$ where $$k_x = k_y = \frac{E A}{L}$$. 2. **Given Data:** - Modulus of Elasticity, $$E = 50 \text{ MPa} = 50 \times 10^6 \text{ Pa}$$ - Foundation dimensions: $$5 \text{ m} \times 5 \text{ m}$$ - Cross-sectional area, $$A = 5 \times 5 = 25 \text{ m}^2$$ - Length, $$L = 5 \text{ m}$$ 3. **Calculate stiffness values:** $$k_x = k_y = \frac{E A}{L} = \frac{50 \times 10^6 \times 25}{5} = 250 \times 10^6 = 2.5 \times 10^8$$ 4. **Matrix K:** $$K = \begin{bmatrix} 2.5 \times 10^8 & 0 \\ 0 & 2.5 \times 10^8 \end{bmatrix}$$ 5. **Eigenvalues:** For a diagonal matrix, eigenvalues are the diagonal elements: $$\lambda_1 = 2.5 \times 10^8, \quad \lambda_2 = 2.5 \times 10^8$$ 6. **Eigenvectors:** Eigenvectors correspond to the standard basis vectors: $$\mathbf{v}_1 = \begin{bmatrix}1 \\ 0\end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix}0 \\ 1\end{bmatrix}$$ **Final answer:** - Eigenvalues: $$2.5 \times 10^8, 2.5 \times 10^8$$ - Eigenvectors: $$\begin{bmatrix}1 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 1\end{bmatrix}$$