Sylvester's law of inertia

Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if S {\displaystyle S} is a symmetric matrix, then for any invertible matrix P {\displaystyle P} , the numbers of positive, negative, and zero eigenvalues of S ′ = P S P T {\displaystyle S'=PSP^{\mathsf {T}}} are constant (i.e., the inertia of S ′ {\displaystyle S'} is constant).

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