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. Eigenvalues of a real matrix may be real or appear in
complex pairs.
A nonzero vector satisfying
is a (right) eigenvector
for the eigenvalue
.
A nonzero vector
satisfying
is a left eigenvector
for the eigenvalue
.
An by
matrix need not have
independent eigenvectors.
The simplest example is
Since the eigenvalues may be complex, there is no fixed way to order them.
Nonetheless, it is convenient to number them as
,
with corresponding right eigenvectors
and left eigenvectors
(if they exist).