# Positive Definite Matrix The main point of a positive definite matrix is: > It does not flip eigenvectors or sends them to zero. It only scales them (in positive direction). Put another way: > This also means that orientations are preserved, i.e. the matrix only stretches and compresses things without flipping them. ### Formal Defn The formal definition (that captures the above intuition) is that: $-\frac{\pi}{2} < \langle x, Ax \rangle < \frac{\pi}{2}$ Which is nicely visualized below:  This is just saying that the transformed $x$, i.e. $Ax$, must have an angle of no more than $|\frac{\pi}{2}|$ with $x$. If it does, then clearly our **orientation** has been **flipped**. References: * [Simple definition of a positive definite matrix](https://math.stackexchange.com/questions/1535817/simple-definition-of-a-positive-definite-matrix)