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 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