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
$$K=\frac{1+\sin\theta}{1-\sin\theta}\Leftrightarrow\cos\theta=\frac{2\sqrt{K}}{{1+K}}.$$
For $n=2$ this reduces to the first equation in the OP.

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I just noticed a similar answer at <A HREF="https://math.stackexchange.com/questions/2266057/maximum-angle-between-a-vector-x-and-its-linear-transformation-a-x">MSE.</A>