Similar matrices and field extensions

Given a field F and a subfield K of F. Let A, B be n×n matrices such that all the entries of A and B are in K. Is it true that if A is similar to B in Fn×n then they are similar in Kn×n?

Any help … thanks!

Answer

If the fields are infinite, there is an easy proof.

Let FK be a field extension with F infinite. Let A,BMatn(F) be two square matrices that are similar over K. So there is a matrix MGLn(K) such that AM=MB. We can write:
M=M1e1++Mrer,
with MiMn(F) and {e1,,er} is a F-linearly independent subset of K. So we have AMi=MiB for every i=1,,r. Consider the polynomial
P(t1,,tr)=det(t1M1++trMr)F[t1,,tr].
Since detM0, P(e1,,er)0, hence P is not the zero polynomial. Since F is infinite, there exist λ1,,λrF such that P(λ1,,λr)0. Picking N=λ1M1++λrMr, we have N \in \mathrm{GL}_n(F) and A N = N B.

Attribution
Source : Link , Question Author : Melesia , Answer Author : Andrea

Leave a Comment