Since similarity of matrices' is an equivalence relation, doesn't that imply that given any polynomial equation involving similar matrices you can substitute in any similar matrices' and the equation will still hold?
For example, given $A,B,C\in M^F_{n\times n}$
if $B \cong C$ then: $$A \cong B^2+5B+3I \iff A \cong C^2+5C+3I$$
Correct?
Best Answer
What you say is correct (although it doesn't follow just from the fact that similarity is an equivalence relation, but from the fact it is an equivalence relation preserved by polynomials, that is if $A\cong B$, then for any polynomial $p$, $p(A)\cong p(B)$).
If $B=PCP^{-1}$ and for some polynomial $p$, $p(B)=DAD^{-1}$, then $p(B)=p(PCP^{-1})=Pp(C)P^{-1}$, so $p(C)=(P^{-1}D)A(D^{-1}P)=(P^{-1}D)A(P^{-1}D)^{-1}$