With this in mind, the one-to-one change of variable formula_136 shows that formula_73 is real and positive for any complex vector formula_74 if and only if formula_139 is real and positive for any formula_140; in other words, if formula_132 is positive definite. For a diagonal matrix, this is true only if each element of the main diagonal—that is, every eigenvalue of formula_2—is positive. Since the spectral theorem guarantees all eigenvalues of a Hermitian matrix to be real, the positivity of eigenvalues can be checked using Descartes' rule of alternating signs when the characteristic polynomial of a real, symmetric matrix formula_2 is available.