Spectral decomposition of a symmetric matrix

Orthonormal eigenbasis. Since $A$ is symmetric, there is an orthonormal basis $(u_1,\ldots,u_n)$ of eigenvectors, with $Au_j=\mu_j u_j$ (where $\mu_j$ ranges over the $\lambda_i$).

Projections. $P_i$ is the orthogonal projection onto $E_{\lambda_i}$: for any $v$, $P_i v$ is the sum of the components of $v$ along eigenvectors with eigenvalue $\lambda_i$.

Spectral decomposition. For any $v$:
$$Av=\sum_j \mu_j\langle v,u_j\rangle u_j=\sum_i \lambda_i\sum_{j:\mu_j=\lambda_i}\langle v,u_j\rangle u_j=\sum_i\lambda_i P_i v.$$
So $A=\sum_i\lambda_i P_i$.

Completeness. $\sum_i P_i$ projects onto the sum of the eigenspaces, which is $\mathbb{R}^n$, so $\sum_i P_i=I$.

Conclusion. Symmetric matrices admit a spectral decomposition as a sum of orthogonal projections scaled by eigenvalues.