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As explained in these notes, 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 $$K=\frac{1+\sin\theta}{1-\sin\theta}\Leftrightarrow\cos\theta=\frac{2\sqrt{K}}{{1+K}}.$$