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 ${\displaystyle S}$ is a symmetric matrix, then for any invertible matrix ${\displaystyle P}$, the numbers of positive, negative, and zero eigenvalues of ${\displaystyle S'=PSP^{\mathsf {T}}}$ are constant (i.e., the inertia of ${\displaystyle S'}$ is constant). This result is particularly useful when ${\displaystyle S'}$ is diagonal, as the inertia of a diagonal matrix can easily be obtained by looking at the signs of its diagonal elements.

This property is named after James Joseph Sylvester, who published its proof in 1852.