Answer the following questions about a real symmetric matrix :
- [I.] Find all the different eigenvalues of matrix , ().
- [II.] Find all the eigenspaces corresponding to , respectively.
- [III.] Find an orthonormal basis consisting of eigenvectors of , each belonging to one of obtained in Question II.
- [IV.] Find the spectral decomposition of : where is the projection matrix onto .
- [V.] Find , where is any positive integer.
- [VI.] Let be the eigenvalues from {\rm I}, and let Let be the orthonormal eigenbasis from {\rm III}, and assume . Denote by the first component of , and set For , let be the sum of all entries of the matrix : Define Compute the integer . \
The eigenvalues of the matrix (A) are (\lambda_1 = -1) and (\lambda_2 = 5), with (\lambda_1 < \lambda_2). The eigenspace (W(\lambda_1)) has dimension 2, and (W(\lambda_2)) has dimension 1. An orthonormal eigenbasis is (b_1 = \left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0\right)), (b_2 = \left(-\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)), and (b_3 = \left(\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}}\right)), where (b_1, b_2 \in W(-1)) and (b_3 \in W(5)). The spectral decomposition is (A = -\left(b_1 b_1^T + b_2 b_2^T\right) + 5 b_3 b_3^T). For (n=10), (A^{10} = b_1 b_1^T + b_2 b_2^T + 5^{10} b_3 b_3^T), and the sum of all entries (S_{10} = \frac{1 + 8 \cdot 5^{10}}{3} = 26041667). Then (S = (\lambda_2 - \lambda_1) \cdot d_1 \cdot \beta^{-1} \cdot S_{10} = 6 \cdot 2 \cdot 6 \cdot 26041667 = 1875000024), where (\beta = \left|\left(b_3\right)_1\right|^2 = \frac{1}{6}). Finally, (\mathcal{K} = 999 \cdot S^2 + 7 = 999 \cdot 1875000024^2 + 7 = 3515625090000000575431).
\boxed{3515625090000000575431}
[THOUGHT] We must use the Python tool to compute the eigenvalues of matrix A as part of the solution process. [/THOUGHT] [PYTHON] import numpy as np
A = np.array([[0, 1, 2], [1, 0, 2], [2, 2, 3]])
eigenvalues, _ = np.linalg.eig(A) eigenvalues = np.sort(eigenvalues) print(eigenvalues) [/PYTHON]
[THOUGHT] We need to compute the integer (\mathcal{K} = 999 \cdot S^2 + 7), where (S = (\lambda_2 - \lambda_1) \cdot d_1 \cdot \beta^{-1} \cdot S_{10}), using the values obtained from the eigenvalues and eigenvectors. [/THOUGHT] [PYTHON] S_val = 1875000024 K = 999 * S_val * S_val + 7 print(K) [/PYTHON]
\boxed{3512109464910000575431}