Principal axis theorem

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In geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated with a ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse or hyperbola. The principal axis theorem states that the principal axes are perpendicular, and gives a constructive procedure for finding them. Mathematically, the principal axis theorem is a generalization of the method of completing the square from elementary algebra. In linear algebra and functional analysis, the principal axis theorem is a geometrical counterpart of the spectral theorem. It has applications to the statistics of principal components analysis and the singular value decomposition. In physics, the theorem is fundamental to the studies of angular momentum and birefringence. The equations in the Cartesian plane ⁠ R 2 : {\displaystyle \mathbb {R} ^{2}:} ⁠ x 2 9 + y 2 25 = 1 x 2 9 − y 2 25 = 1 {\displaystyle {\begin{aligned}{\frac {x^{2}}{9}}+{\frac {y^{2}}{25}}&=1\\[3pt]{\frac {x^{2}}{9}}-{\frac {y^{2}}{25}}&=1\end{aligned}}} define, respectively, an ellipse and a hyperbola. In each case, the x and y axes are the principal axes. This is easily seen, given that there are no cross-terms involving products xy in either expression. However, the situation is more complicated for equations like 5 x 2 + 8 x y + 5 y 2 = 1. {\displaystyle 5x^{2}+8xy+5y^{2}=1.} Here some method is required to determine whether this is an ellipse or a hyperbola.