# Positive Definite Matrices Motivation: Verifying that a probability matrix in the [[MLEM for Compton Imaging]] is positive definite. A matrix is considered positive definite if it is symmetric, and all of its eigenvalues $\lambda$ are positive $(\lambda > 0)$ [^1]. It also follows that if a matrix $A$ is positive definite, then it is invertible and $det(A) > 0$ [^1]:https://math.emory.edu/~lchen41/teaching/2020_Fall/Section_8-3.pdf Another useful theorem is that a symmetric matrix $A$ is positive definite if and only if $\textbf{x}^T A\textbf{x} > 0$ for every column $\textbf{x} \neq \textbf{0}$ in $\mathbb{R}$.