Let $(V,(\cdot,\cdot))$ be a finite-dimensional real Euclidean [vector space](/page/Vector%20Space), let $\Phi \subset V$ be a finite reduced crystallographic root system, and let $\Delta = \{\alpha_1,\dots,\alpha_n\}$ be an ordered simple system for $\Phi$. Let $A = (a_{ij}) \in \mathbb{R}^{n \times n}$ be the Cartan matrix associated to $\Delta$, defined by

\begin{align*} a_{ij} = \frac{2(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)} \end{align*}

for $1 \leq i,j \leq n$. Then there exists a diagonal matrix $D \in \mathbb{R}^{n \times n}$ with positive real diagonal entries such that $DA$ is symmetric positive definite.