Two standard approaches to the computation of eigenvectors corresponding to eigenvalues of a non-symmetric real matrix are (1) a backward substitution on the real Schur matrix followed by a matrix-matrix multiplication and (2) inverse iteration on the Hessenberg matrix. Both approaches have been realized as tiled, task-parallel solvers where a large proportion of the operations are matrix-matrix multiplications. The first contribution of this work addresses a memory bottleneck of the inverse iteration solver by improving the locality of memory accesses. The second contribution compares the two approaches quantitatively with numerical experiments.