Lyapunov exponent
and by virtue of stretching, rotating and Gram-Schmidt orthogonalization the Lyapunov exponents are ordered as formula_26. When iterated forward in time a random vector contained in the space spanned by the first formula_20 columns of formula_19 will almost surely asymptotically grow with the largest Lyapunov exponent and align with the corresponding Lyapunov vector. In particular, the first column of formula_19 will point in the direction of the Lyapunov vector with the largest Lyapunov exponent if formula_30 is large enough. When iterated backward in time a random vector contained in the space spanned by the first formula_20 columns of formula_32 will almost surely, asymptotically align with the Lyapunov vector corresponding to the formula_20th largest Lyapunov exponent, if formula_30 and formula_35 are sufficiently large. Defining formula_36 we find formula_37. Choosing the first formula_20 entries of formula_39 randomly and the other entries zero, and iterating this vector back in time, the vector formula_40 aligns almost surely with the Lyapunov vector formula_41 corresponding to the formula_20th largest Lyapunov exponent if formula_35 and formula_30 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.