Very basic question about Wiedemann methods

[fivemack]

I can see (and have checked by experiment in python and written up in wikipedia) roughly how the Coppersmith algorithm works - we know that there exists a minimal polynomial for the matrix which will annihilate the sequence y M^i x, we know a polynomial that annihilates the sequence y M^i x, we can use the second as if it were the first and generally err only by some small polynomial factor.

But I can't understand how I can obtain a vector annihilated by a matrix if I know the minimal polynomial of the matrix.