  This paper proves the global convergence of a triangularized
orthogonalization-free method (TriOFM). TriOFM, in general, applies a
triangularization idea to the gradient of an objective function and removes the
rotation invariance in minimizers. More precisely, in this paper, the TriOFM
works as an eigensolver for sizeable sparse matrices and obtains eigenvectors
without any orthogonalization step. Due to the triangularization, the iteration
is a discrete-time flow in a non-conservative vector field. The global
convergence relies on the stable manifold theorem, whereas the convergence to
stationary points is proved in detail in this paper. We provide two proofs
inspired by the noisy power method and the noisy optimization method,
respectively.
