An exceptional point, in the branch-point terminology for parameter-dependent spectra, is a branch point at which eigenvalues and their associated eigenvectors coalesce. For a matrix family, the usage here additionally requires the coalescing eigenvalue to have geometric multiplicity smaller than its algebraic multiplicity, making the matrix at that point a defective matrix (Heiss 2004, 2012).
A simple example is the analytic matrix family
![A(z)=[0 1; z 0],](/images/equations/ExceptionalPoint/NumberedEquation1.svg) |
(1)
|
where 
. Its eigenvalues
are
 |
(2)
|
For 
, corresponding eigenvectors
are 
, where 
denotes the transpose. At 
, both eigenvalues
vanish and both eigenvectors approach 
. There is only one independent eigenvector,
and 
is a nontrivial block in Jordan canonical form.
Continuing around the origin once exchanges the two branches of 
, illustrating the monodromy
of an algebraic branch point.
Since 
, the matrix
exponential is
 |
(3)
|
The extra factor of 
is a consequence of the nontrivial block in Jordan
canonical form. For real 
, acting on 
gives 
, whose squared vector
norm is 
despite the matrix having only the zero eigenvalue.
Defectiveness alone does not ensure branching for a specified parameterization. Replacing 
by 
in this example gives analytic eigenvalues 
and 
while leaving the matrix at the
origin defective. The branch-point usage distinguishes
such a crossing from an exceptional point (Heiss 2012). Transitions between real
and nonreal eigenvalues in parity-time
symmetry provide another example.
An avoided crossing for a real parameter can reflect branch point structure at complex parameter values (Heiss and Steeb 1991). A diabolic point, in contrast, has two linearly independent eigenvectors at a double eigenvalue (Seyranian et al. 2005, pp. 1723-1725).
