Product Eigenvalue Problems

Many eigenvalue problems are most naturally viewed as product eigenvalue problems. The eigenvalues of a matrix A are wanted, but A is not given explicitly. Instead it is presented as a product of several factors: $A=A_{k}A_{k-1}\cdots A_{1}$. Usually more accurate results are obtained by working with the factors rather than forming A explicitly. For example, if we want eigenvalues/vectors of $B^{T}B$, it is better to work directly with B and not compute the product. The intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept. Diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems. For all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic GR algorithm applied to a related cyclic matrix.