The numbers of eigenvalues that are positive, negative, or 0 do not change under a congruence transformation.
Gradshteyn and Ryzhik (2000) state it as follows: when a quadratic
form 
in 
variables is reduced by a nonsingular linear transformation to the form
 |
the number 
of positive squares
appearing in the reduction is an invariant of the quadratic
form 
and does not depend on the method of reduction.
