As explained in these <A HREF="https://calculus7.org/2013/05/04/condition-number-and-maximal-rotation-angle/">notes</A>, the maximum rotation angle $\theta$ of a symmetric positive definite matrix $M$ is related to the condition number $K=\mu_{\rm max}/\mu_{\rm min}$ of the matrix (the ratio of largest and smallest eigenvalue) by
$$K=\frac{1+\sin\theta}{1-\sin\theta}\Leftrightarrow\cos\theta=\frac{2\sqrt{K}}{{1+K}}.$$