Decompositions and coalescing eigenvalues of symmetric definite pencils depending on parameters

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) − λB(x)) v = 0, where A and B are real symmetric matrix valued functions of order n, smoothly depending on two real parameters x = (x1, x2); 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). The main goal of the paper is precisely that of locating parameter values where eigenvalues are equal. We first give general theoretical results for the present generalized eigenvalue problem, and then introduce and implement numerical methods apt at detecting conical intersections. Finally, we 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.