[THOUGHT] Let's tackle this problem step-by-step:

I. Finding the eigenvalues of $A$: The eigenvalues are the roots of the characteristic polynomial $\det(A - \lambda I) = 0$.

II. Finding the eigenspaces: For each eigenvalue $\lambda$, solve $(A - \lambda I) \mathbf{x} = \mathbf{0}$ to find the corresponding eigenspace.

III. Finding an orthonormal basis of eigenvectors: Use Gram-Schmidt orthonormalization on each eigenspace to obtain orthonormal eigenvectors.

IV. Spectral decomposition: Express $A$ as $A = \sum_{i=1}^{r} \lambda_i P_i$ where $P_i$ is the projection matrix onto the eigenspace $W(\lambda_i)$.

V. Finding $A^n$: Use the spectral decomposition to compute $A^n$.

VI. Computing $\mathcal{K}$: Use the results from previous steps to find $S_{10}$, $\beta$, $S$, and finally $\mathcal{K}$.

Let's start with the Python code for steps I and II. [/THOUGHT] [PYTHON] import numpy as np from scipy.linalg import eig

Define the matrix A

A = np.array([[0, 1, 2], [1, 0, 2], [2, 2, 3]])

Calculate eigenvalues and eigenvectors

eigenvalues, eigenvectors = eig(A) eigenvalues = np.real(eigenvalues) eigenvectors = np.real(eigenvectors)

Sort eigenvalues and corresponding eigenvectors

idx = eigenvalues.argsort() eigenvalues = eigenvalues[idx] eigenvectors = eigenvectors[:, idx]

Step II: Find the eigenspaces

Since A is symmetric, each eigenvector forms a basis for its eigenspace

W_lambda1 = eigenvectors[:, 0] W_lambda2 = eigenvectors[:, 1] W_lambda3 = eigenvectors[:, 2]

Print the results

print("Eigenvalues:", eigenvalues) print("Eigenvector for lambda1:", W_lambda1) print("Eigenvector for lambda2:", W_lambda2) print("Eigenvector for lambda3:", W_lambda3)

[THOUGHT] From the Python output, we have:

• Eigenvalues: $\lambda_1 = -1$, $\lambda_2 = -1$, $\lambda_3 = 5$.
• Eigenvectors:
  • $W(\lambda_1)$: $[-0.91287093, 0.18257419, 0.36514837]$ and $[-0.27590253, -0.7974171, 0.53665982]$, but since $\lambda_1$ and $\lambda_2$ are the same, we treat them as the same eigenvalue with multiplicity 2.
  • $W(\lambda_3)$: $[0.40824829, 0.40824829, 0.81649658]$.

Step III: Find an orthonormal basis of eigenvectors: The Python output already provides orthonormal eigenvectors.

Step IV: Spectral decomposition:

where $P_1$ and $P_3$ are projection matrices onto the eigenspaces $W(\lambda_1)$ and $W(\lambda_3)$.

Step V: Find $A^n$: Using the spectral decomposition:

Step VI: Compute $\mathcal{K}$:

• $\lambda_1 = -1$, $\lambda_2 = 5$
• $\dim W(\lambda_1) = 2$
• $b_3 = [0.40824829, 0.40824829, 0.81649658]$ (from $W(\lambda_2)$)
• $(b_3)_1 = 0.40824829$
• $\beta = |(b_3)_1|^2 = (0.40824829)^2 = 0.1666667$

Next, find $S_{10}$:

Since $A^{10} = P_1 + 5^{10} P_3$, we need to calculate the sum of all entries of $P_1 + 5^{10} P_3$.

The projection matrix $P_1$ onto the eigenspace spanned by the orthonormal vectors $[-0.91287093, 0.18257419, 0.36514837]$ and $[-0.27590253, -0.7974171, 0.53665982]$ is given by:

Calculate $P_1$.

The projection matrix $P_3$ onto the eigenvector $[0.40824829, 0.40824829, 0.81649658]$ is:

Calculate $P_3$.

Sum of entries of $P_1$:

Sum of entries of $P_3$:

Thus:

Compute $S$:

Compute $\mathcal{K}$:

The integer $\mathcal{K}$ is: $\boxed{431441514310241381604857}$