  M\"{o}bius transformations have been used in numerical algorithms for
computing eigenvalues and invariant subspaces of structured generalized and
polynomial eigenvalue problems (PEPs). These transformations convert problems
with certain structures arising in applications into problems with other
structures and whose eigenvalues and invariant subspaces are easily related to
the ones of the original problem. Thus, an algorithm that is efficient and
stable for some particular structure can be used for solving efficiently
another type of structured problem via an adequate M\"{o}bius transformation. A
key question in this context is whether these transformations may change
significantly the conditioning of the problem and the backward errors of the
computed solutions, since, in that case, their use may lead to unreliable
results. We present the first general study on the effect of M\"{o}bius
transformations on the eigenvalue condition numbers and backward errors of
approximate eigenpairs of PEPs. By using the homogeneous formulation of PEPs,
we are able to obtain two clear and simple results. First, we show that, if the
matrix inducing the M\"{o}bius transformation is well conditioned, then such
transformation approximately preserves the eigenvalue condition numbers and
backward errors when they are defined with respect to perturbations of the
matrix polynomial which are small relative to the norm of the polynomial.
However, if the perturbations in each coefficient of the matrix polynomial are
small relative to the norm of that coefficient, then the corresponding
eigenvalue condition numbers and backward errors are preserved approximately by
the M\"{o}bius transformations induced by well-conditioned matrices only if a
penalty factor, depending on those coefficients, is moderate. It is important
to note that these simple results are no longer true if a non-homogeneous
formulation is used.
