How to do spectral decomposition?

Question

I missed the last couple classes due to a family emergency and am trying to catch up with review questions. However, I can't seem to find an online source that teaches how to compute a spectral decomposition. I would really appreciate it if someone here can explain to me how the following problem would be done (at the level of someone just learning linear algebra). Thank you!

5. Compute the spectral decomposition of $ \begin{bmatrix} -2 & 6 \\ 6 & 7 \end{bmatrix} $ and $ \begin{bmatrix} 1 & 0 & 2 \\ 0 & -1 & -2 \\ 2 & -2 & 0 \end{bmatrix}. $

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Answer 1

Given a symmetric matrix $A \in M_n(\mathbb{R})$, what is usually meant by finding the spectral decomposition of $A$ is to find an orthogonal matrix $O \in M_n(\mathbb{R})$ and a diagonal matrix $D \in M_n(\mathbb{R})$ such that $O^T A O = D$. In order to find $D$ and $O$, you can apply the following steps:

1. Find the eigenvalues of $A$ - they will be the roots of the characteristic polynomial $p_A(\lambda) = \det(\lambda I - A)$. Call them $\lambda_1, \ldots, \lambda_k$.
2. For each eigenvalue $\lambda_i$ of $A$, find a basis of the corresponding eigenspace $V_{\lambda_i} = \{ v \in \mathbb{R}^n \, | \, Av = \lambda_i v \} = \mathrm{span} \{ v_1^i, \ldots, v_{m_i}^i \}$. This is done by solving a system of linear equations.
3. By applying the Gram-Schmidt procedure if necessarily, replace each basis $\{v^i_1, \ldots, v^i_{m_i} \}$ with an orthonormal basis $\{ w^i_1, \ldots, w_{m_i}^i \}$ for $V_{\lambda_i}$.
4. Construct a matrix $O$ by taking its columns to be the orthonormal bases for the eigenspaces. That is, let $$ O = (v_1^1 | \ldots | v_{m_1}^1 | v_1^2 | \ldots | v_{m_2}^2 | \ldots | v_1^k | \ldots | v_{m_k}^k ). $$ The matrix $O$ will be orthogonal and $$ O^T A O = \mathrm{diag}(\lambda_1, \ldots, \lambda_1, \lambda_2, \ldots, \lambda_2, \ldots, \lambda_k, \ldots, \lambda_k) = D$$ where each eigenvalue $\lambda_i$ appears $m_i$ times in $D$.