Formula for the spectral decomposition for a symmetric matrix

Spectral Decomposition Formula. If $A$ is a real symmetric matrix with distinct eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_k$ and corresponding orthonormal eigenvectors $v_1, v_2, \ldots, v_k$, then:
$$A = \lambda_1 v_1 v_1^T + \lambda_2 v_2 v_2^T + \cdots + \lambda_k v_k v_k^T$$

Each term $P_i = v_i v_i^T$ is the orthogonal projection onto the eigenspace of $\lambda_i$. This gives $A = \sum_{i=1}^k \lambda_i P_i$.

Example. For $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ with $\lambda_1 = 3, v_1 = \frac{1}{\sqrt{2}}(1,1)^T$ and $\lambda_2 = 1, v_2 = \frac{1}{\sqrt{2}}(1,-1)^T$:
$$A = 3 \cdot \frac{1}{2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} + 1 \cdot \frac{1}{2}\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}$$