[GAP Forum] Frobenius normal form (explicit change of basis)

[Bors Alexander]

Dear GAP Forum,



I am looking for a function that takes as input a square matrix M over a finite field k and outputs a regular matrix T over k, of the same dimension as M, such that TMT^(-1) is in Frobenius normal form (aka rational canonical form). Is there a simple way to construct such a function from GAP's built-in functions?



I am aware that the Frobenius normal form of M per se (i.e., the elementary divisors of M) can be determined by computing the Smith normal form of the matrix M-X*Id (X an indeterminate) and that GAP's function ElementaryDivisorsTransformationsMat can be used to find corresponding left and right transformation matrices P,Q over k[X]. Unfortunately, I don't see how, if at all, P and Q relate to T.



Thank you in advance,

Alexander