Conditioning and backward errors of eigenvalues of homogeneous matrix polynomials under Möbius transformations

Abstract

We present the first general study on the effect of Möbius transformations on the eigenvalue condition numbers and backward errors of approximate eigenpairs of polynomial eigenvalue problems (PEPs). By usingthe homogeneous formulation of PEPs, we are able to obtain two clear andsimple results. First, we show that if the matrix inducing the Möbius transformation is well-conditioned, then such transformation approximately preservesthe eigenvalue condition numbers and backward errors when they are definedwith respect to perturbations of the matrix polynomial which are small relativeto the norm of the whole polynomial. However, if the perturbations in eachcoefficient of the matrix polynomial are small relative to the norm of that coefficient, then the corresponding eigenvalue condition numbers and backwarderrors are preserved approximately by the Möbius transformations induced bywell-conditioned matrices only if a penalty factor, depending on the norms ofthose matrix coefficients, is moderate. It is important to note that these simple results are no longer true if a non-homogeneous formulation of the PEP isused.