rates and by virtue of stretching, rotating and Gram-Schmidt orthogonalization the Lyapunov exponents are ordered as . When iterated forward in time a random vector contained in the space spanned by the first columns of will almost surely asymptotically grow with the largest Lyapunov exponent and align with the corresponding Lyapunov vector. In particular, the first column of will point in the direction of the Lyapunov vector with the largest Lyapunov exponent if is large enough. When iterated backward in time a random vector contained in the space spanned by the first columns of will almost surely, asymptotically align with the Lyapunov vector corresponding to the th largest Lyapunov exponent, if and are sufficiently large. Defining we find . Choosing the first entries of randomly and the other entries zero, and iterating this vector back in time, the vector aligns almost surely with the Lyapunov vector corresponding to the th largest Lyapunov exponent if and are sufficiently large. Since the iterations will exponentially blow up or shrink a vector it can be re-normalized at any iteration point without changing the direction. References Functional analysis Mathematical physics Dynamical systems