If 
is real, symmetric, i.e., 
or
, (where 
is the conjugate transpose), and the
eigenvalues of , 
,
are not necessarily distinct but the associated eigenvectors
still form a basis, then is called simple and the
diagonal form for exists.
If eigenvectors do not form a basis,
, then the diagonal form DNE. Instead we have a Jordan
canonical form.
Let 
, 
, 
, 
.
, and 
is
called a Jordan block of dimension 
. Usually
, and 
. Then
Suppose 
then 
so
and 
are called generalized eigenvectors.
Example 
.
so 
is an eigenvalue of multiplicity 
.
therefore there is only 
eigenvector
Then
Same method gives
So
Check:
