  In this work, we consider symmetric positive definite pencils depending on
two parameters. That is, we are concerned with the generalized eigenvalue
problem $A(x)-\lambda B(x)$, where $A$ and $B$ are symmetric matrix valued
functions in ${\mathbb R}^{n\times n}$, smoothly depending on parameters $x\in
\Omega\subset {\mathbb R}^2$; further, $B$ is also positive definite. In
general, the eigenvalues of this multiparameter problem will not be smooth, the
lack of smoothness resulting from eigenvalues being equal at some parameter
values (conical intersections). We first give general theoretical results on
the smoothness of eigenvalues and eigenvectors for the present generalized
eigenvalue problem, and hence for the corresponding projections, and then
perform a numerical study of the statistical properties of coalescing
eigenvalues for pencils where $A$ and $B$ are either full or banded, for
several bandwidths. Our numerical study will be performed with respect to a
random matrix ensemble which respects the underlying engineering problems
motivating our study.
