A Frank matrix is an upper Hessenberg matrix whose entries, in one standard convention, are
 |
(1)
|
for 
(Frank 1958; Higham 1995,
p. 43). Its determinant is 1, and its eigenvalues
are positive and occur in reciprocal pairs. The smaller eigenvalues
can be highly sensitive to perturbations (Higham 1995, pp. 26-29).
The plot shows absolute condition numbers of the eigenvalues for 
, ordered from smallest to largest. For right and left eigenvectors 
and 
of a simple eigenvalue, the quantity plotted is 
, where 
denotes the conjugate transpose.
