Sylvester's inertia law states that the numbers of positive, negative, and 0 eigenvalues
of a real symmetric matrix do not change
under a nonsingular congruence transformation.
Gradshteyn and Ryzhik (2000) state it as follows: when a real quadratic
form 
in 
variables is reduced by a nonsingular linear
transformation to the form
 |
(1)
|
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.
The law also counts generalized eigenvalues. Let 
and 
be real symmetric matrices, with 
a positive definite
matrix, and let 
, 
, ..., 
be the generalized
eigenvalues of 
, counted with multiplicity.
If 
denotes the number of negative eigenvalues
of a real symmetric matrix 
, then for every real 
,
 |
(2)
|
Indeed, 
is related by a congruence transformation
to 
,
whose eigenvalues are 
.
A symmetric factorization
 |
(3)
|
with 
nonsingular and 
a block diagonal matrix with symmetric 
and 
blocks, gives 
. The latter count is obtained from the
eigenvalues of the individual blocks. Thus, if 
and neither endpoint is
a generalized eigenvalue, the difference
counts the generalized eigenvalues in 
, including multiplicities.
Such counts are used to check that a computation has found all generalized
eigenvalues in an interval (Grimes et al. 1994).
