= Principal axes of an ellipse
{title2=$(\mathbf x-\mathbf c)^TM(\mathbf x-\mathbf c)=1$}
The <principal axes of an ellipse> are the two perpendicular symmetry axes through its <center of an ellipse>. For a <symmetric positive-definite matrix> $M$, its orthonormal <eigenvectors> give those directions and its positive <eigenvalues> $\lambda_1,\lambda_2$ give semiaxis lengths $\lambda_1^{-1/2},\lambda_2^{-1/2}$. This follows by <orthogonal diagonalization> of the defining <quadratic form>. In $\xi^2/A^2+\upsilon^2/B^2=1$, the axes are already horizontal and vertical, with semiaxes $|A|,|B|$ and full lengths $2|A|,2|B|$. Equal semiaxes give a <circle>, for which every perpendicular pair through the center can serve.
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