A Forsythe matrix is a Jordan block with a perturbation in its lower left corner (Higham 1995, p. 43). For 
, it is
 |
where 
has 
on the main diagonal and ones on the superdiagonal,
is the 
th standard basis vector,
and 
denotes the transpose. Its characteristic
polynomial is
 |
For 
,
its eigenvalues lie on a circle
of radius 
centered at 
. A very small change in a single entry can therefore
produce a much larger displacement of eigenvalues
from those of the unperturbed Jordan block.
The plot uses 
,
, and 
(green), 
(orange), and 
(blue).
