Problem Analysis #7

Answer the following questions about a real symmetric matrix $A$:$A =\begin{pmatrix}0 & 1 & 2 \\1 & 0 & 2 \\2 & 2 & 3\end{pmatrix}.$

• [I.] Find all the different eigenvalues of matrix $A$, $\lambda_1, \cdots, \lambda_r$ ($\lambda_1 < \cdots < \lambda_r$).
• [II.] Find all the eigenspaces $W(\lambda_1), \cdots, W(\lambda_r)$ corresponding to $\lambda_1, \cdots, \lambda_r$, respectively.
• [III.] Find an orthonormal basis $b_1, b_2, b_3$ consisting of eigenvectors of $A$, each belonging to one of $W(\lambda_1), \cdots, W(\lambda_r)$ obtained in Question II.
• [IV.] Find the spectral decomposition of $A$: $A = \sum_{i=1}^{r} \lambda_i P_i,$ where $P_i$ is the projection matrix onto $W(\lambda_i)$.
• [V.] Find $A^n$, where $n$ is any positive integer.
• [VI.] Let $\lambda_1 < \lambda_2$ be the eigenvalues from {\rm I}, and let $d_1 := \dim W(\lambda_1).$ Let $b_1,b_2,b_3$ be the orthonormal eigenbasis from {\rm III}, and assume $b_3 \in W(\lambda_2)$. Denote by $(b_3)_1$ the first component of $b_3$, and set $\beta := |(b_3)_1|^2.$ For $n=10$, let $S_{10}$ be the sum of all entries of the matrix $A^{10}$: $S_{10} := \sum_{i=1}^3 \sum_{j=1}^3 (A^{10})_{ij}.$ Define $S := (\lambda_2 - \lambda_1)\, d_1\, \beta^{-1} S_{10}, \qquad \mathcal{K} := 999\,S^2 + 7.$ Compute the integer $\mathcal{K}$. \