Exact Linear Algebra for SAGE - William Stein - University of ...

Echelon Forms of Matrices
Computing Echelon Forms
Decomposing Spaces Under the Action of Matrix
Wiedemann’s Minimal Algorithm
This gives a flavor for advanced charpoly algorithms.
Choose a random vector v and compute the iterates
v0 = v, v1 = A(v), v2 = A 2 (v), . . . , v2n−1 = A 2n−1 (v). (3.1)
If f = x m + cm−1x m−1 + · · · + c1x + c0 is the minimal polynomial of A, then
A m + cm−1A m−1 + · · · + c0In = 0,
where In is the n × n identity matrix. For any k ≥ 0, by multiplying both sides
on the right by the vector A k v, we see that
hence
A m+k v + cm−1A m−1+k v + · · · + c0A k v = 0,
vm+k + cm−1vm−1+k + · · · + c0vk = 0, all k ≥ 0.
Any single component of the vectors v0, . . . , v2n−1 satisfies the linear
recurrence with coefficients 1, cm−1, . . . , c0. The Berlekamp-Massey
algorithm finds the minimal polynomial of a linear recurrence sequence {ar },
which is a factor of the minimal polynomial of A.
This algorithm is especially good for sparse matrices.
William Stein Exact Linear Algebra for SAGE