Rational canonical form conjugation

Question

I know that if A is an nxn intger matrix, with A = XRX^-1 where R is the rcf of A. Then R is also an integer matrix.

My question is Does X and X^- are integer matrices as well ?

Answer 1

There's no reason for the similarity matrices to be integer matrices. The most natural construction is to start with some fairly arbitrary $v_1$ and define $v_2=Av_1$, $v_3=Av_2$, and so on until we run out of room and they stop being linearly independent; the $v_i$ will be columns of $X$. While starting with an integer vector $v_1$ will make the other $v_i$ integer vectors, there's no control of the determinant, and we will be dividing by something when it comes time to take the inverse $X^{-1}$.

As an example, I constructed a random (entries discrete uniform between $-5$ and $5$) $5\times 5$ matrix: $$A=\begin{pmatrix}4&-4&-2&-3&2\\0&-4&1&-5&-3\\4&5&4&-5&-3\\-3&-3&-2&0&4\\3&-5&3&-5&2\end{pmatrix},\quad X=\begin{pmatrix}1&4&23&162&1749\\0&0&10&23&853\\0&4&38&319&2665\\0&-3&-32&-355&-2725\\0&3&45&383&3869\end{pmatrix}$$ Then $X$ has determinant $183716$, so its inverse certainly isn't an integer matrix.

Incidentally, for this example, we get $R=\begin{pmatrix}0&0&0&0&-1349\\1&0&0&0&1531\\0&1&0&0&-548\\0&0&1&0&71\\0&0&0&1&6\end{pmatrix}$.