Formula for the spectral decomposition for a symmetric matrix

For a real symmetric $A$, choose an orthonormal eigenvector basis $q_1,\ldots,q_n$ with eigenvalues $\lambda_1,\ldots,\lambda_n$, allowing repetitions. Then $A=\sum_{j=1}^n\lambda_jq_jq_j^T$.

Alternatively group **distinct** eigenvalues $\mu$: $A=\sum_\mu\mu P_\mu$, where $P_\mu=\sum_{j:\lambda_j=\mu}q_jq_j^T$ projects onto the entire eigenspace. One eigenvector per distinct eigenvalue is insufficient when an eigenspace has dimension greater than one.

For $\begin{bmatrix}2&1\\1&2\end{bmatrix}$, use $q_1=(1,1)^T/\sqrt2$, $q_2=(1,-1)^T/\sqrt2$ and eigenvalues $3,1$.Copy