The polynomial
is called the
characteristic polynomial
of
. The roots of
are called the 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
.
Since
,
left and right eigenvectors are identical.
All eigenvalues of a Hermitian matrix 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 eigenvalue has an eigenvector
.
We may assume without loss of generality that
.
Each
is real if
is real.
Though the
may not be unique (e.g., any vector is an eigenvector of
the identity matrix), they may be chosen to all be orthogonal to one another:
if
.
When an eigenvalue is distinct from all the other eigenvalues,
its eigenvector is unique (up to multiplication by scalars).