The polynomial
is called the
characteristic polynomial
of
. The roots of
are eigenvalues of
.
Since the degree of
is
, it has
roots, and so
has
eigenvalues.
A nonzero vector satisfying
is a
(right) eigenvector
for the eigenvalue
. The eigenpair
also satisfies
, so we can also call
a left eigenvector.
All eigenvalues of the definite pencil are real.
This lets us write
them in sorted order
.
If all
, then
is called
positive definite,
and if all
, then
is called
positive semidefinite.
Negative definite and negative semidefinite are defined
analogously. If there are both positive and negative eigenvalues,
is called indefinite.
Each is real if
and
are real.
Though the
may not be unique,
they may be chosen to all be
orthogonal to one another:
if
. This is also called orthogonality
with respect
to the inner product induced by the Hermitian positive definite matrix
.
When an eigenvalue is distinct from all the other eigenvalues,
its eigenvector is unique (up to multiplication by scalars).