Sylvester's Inertia Law

The numbers of Eigenvalues that are Positive, Negative, or 0 do not change under a congruence transformation. Gradshteyn and Ryzhik (1979) state it as follows: when a Quadratic Form $Q$ in $n$ variables is reduced by a nonsingular linear transformation to the form

$$Q={y_1}^2+{y_2}^2+\ldots+{y_p}^2-{p_{p+1}}^2-{y_{p_2}}^2-\ldots-{y_r}^2,$$

the number $p$ of Positive Squares appearing in the reduction is an invariant of the Quadratic Form $Q$ and does not depend on the method of reduction.

See also Eigenvalue, Quadratic Form

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1105, 1979.