On the least squares approximation of symmetric-definite pencils subject to generalized spectral constraints

A general framework for the least squares approximation of symmetric-definite pencils subject to generalized eigenvalues constraints is developed in this paper. This approach can be adapted to different applications, including the inverse eigenvalue problem. The idea is based on the observation that a natural parameterization for the set of symmetric-definite pencils with the same generalized eigenvalues is readily available. In terms of these parameters, descent flows on the isospectral surface aimed at reducing the distance to matrices of the desired structure can be derived. These flows can be designed to carry certain other interesting properties and may be integrated numerically.