Show that $A = \sum_i \lambda_i x_i x_i^T$

Question

I came across this formula: $A = \sum_i \lambda_i x_i x_i^T$ where $\lambda_i$ is the i'th eigenvalue and $x_i$ the i'th eigenvector of $A$. As I have no idea how it is called, I was not able to find a proof.

Could someone give a pointer where to find one?

Edit: I have looked at the eigendecomposition, but only found the decomposition $A = Q^T \Lambda Q$.

Answer 1

Indeed a one line calculus shows that $A = Q^T \Lambda Q = \sum_i \lambda_i Q_i Q_i^T$ with $Q_i$ denoting the $i^{th}$ column of $Q$.