From: wcw@math.psu.edu (William C Waterhouse)
Newsgroups: sci.math
Subject: Re: similar matrices
Date: 21 Jul 1998 21:18:22 GMT

In article <6p220m$rbo@bgtnsc02.worldnet.att.net>, 
Fred Richman <richman@fau.edu> writes:
>... 
> If two matrices are similar, then are they similar over the
> field generated by their entries?
>...

Yes. (See, e.g., Jacobson, Basic Algebra I.)  For a square matrix
M you look at  X(I) - M as a matrix over k[X].  Row and
column reduction will reduce it to  Diag(d_1, d_2, d_3,...,d_n)
where each polynomial d_i divides d_{i+1} (and probably most of 
the ones at the start are = 1). The nontrivial d_i are the
invariant factors of the matrix, and they determine it
precisely up to similarity.  Clearly this computation is
a rational one and is unaffected by expansion of the field
k. 

William C. Waterhouse
Penn State