  Periodically driven quantum many-body systems play a central role for our
understanding of nonequilibrium phenomena. For studies of quantum chaos,
thermalization, many-body localization and time crystals, the properties of
eigenvectors and eigenvalues of the unitary evolution operator, and their
scaling with physical system size $L$ are of interest. While for static
systems, powerful methods for the partial diagonalization of the Hamiltonian
were developed, the unitary eigenproblem remains daunting. In this paper, we
introduce a Krylov space diagonalization method to obtain exact eigenpairs of
the unitary Floquet operator with eigenvalue closest to a target on the unit
circle. Our method is based on a complex polynomial spectral transformation
given by the geometric sum, leading to rapid convergence of the Arnoldi
algorithm. We demonstrate that our method is much more efficient than the shift
invert method in terms of both runtime and memory requirements, pushing the
accessible system sizes to the realm of 20 qubits, with Hilbert space
dimensions $\geq 10^6$.
