to identify the axes of symmetry. There is a beautiful result called Sylvester's law of inertia concerning the solutions of the homogeneous equation $\mathbb{x}^TA\mathbb{x}=1$ for symmetric (i.e. real hermitian ) matrices $A$, which is a quadratic form and can be extended to nonhomogeneous equations $\mathbb{x}^TA\mathbb{x}+\mathbb{b}^T\mathbb{x}=1$ (not discussed here). By homogeneous, we