Show that A^(-1)BA can never be diagonal.

> A:=Matrix([[a,b],[c,d]]);

> B:=Matrix([[p,q],[r,s]]);

> A^(-1).B.A;

> expand((-d*p/(-a*d+b*c)+b*r/(-a*d+b*c))*b+(-d*q/(-a*d+b*c)+b*s/(-a*d+b*c))*d);

> expand((c*p/(-a*d+b*c)-a*r/(-a*d+b*c))*a+(c*q/(-a*d+b*c)-a*s/(-a*d+b*c))*c);

Since -bdp+b^2r-d^2q+dbs=0

acp-a^2r+c^2q-cas=0 only when a=b=c=d=0 but it contridit with A<>0

So A^(-1)BA can never be diagonal