A diabolic point, in the terminology of matrix perturbation theory, is a parameter value where a double eigenvalue has two linearly independent eigenvectors (Seyranian et al. 2005, pp. 1723-1725). The coincident eigenvalue has algebraic multiplicity and geometric multiplicity both equal to 2. This differs from an exceptional point, where the matrix is defective and the eigenvalue branches meet at a branch point.
The alternative name diabolical point emphasizes the double cone formed by the eigenvalue surfaces near a generic degeneracy of a real symmetric matrix family with two parameters (Berry and Wilkinson 1984, p. 15). For example,
![A(x,y)=[x y; y -x],](/images/equations/DiabolicPoint/NumberedEquation1.svg) |
where 
,
has eigenvalues
 |
At the origin, 
is the zero matrix, and
its eigenspace has dimension
2. The surfaces 
form a double
cone. Fixing 
gives an avoided crossing as 
varies.
Having two linearly independent eigenvectors at the double eigenvalue does not by itself imply
a conical shape. For example, the diagonal matrix
with entries 
and 
also has a double eigenvalue with two independent
eigenvectors at the origin,
but its eigenvalue surfaces
are elliptic paraboloids.
