Suppose is a nonsingular matrix.
Let
and
.
We say that the pencil
is
congruent to
, and that
is a congruence transformation.
If
and
are Hermitian, with
positive definite,
than
and
have these same properties.
Furthermore,
and
have the same eigenvalues, and if
is an eigenvector
of
, so that
, then
is an eigenvector of
.