I know that given any two similar matrices $A,B\in M^F_{n\times n}$ and any two invertible matrices $P,Q$ of the same order then $P^{-1}AP$ is similar to $Q^{-1}BQ$.
However I don't completely understand why. Could someone explain it?
I know that given any two similar matrices $A,B\in M^F_{n\times n}$ and any two invertible matrices $P,Q$ of the same order then $P^{-1}AP$ is similar to $Q^{-1}BQ$.
However I don't completely understand why. Could someone explain it?
We use the fact that if $A_1$ is similar to $A_2$ and $A_2$ to $A_3$ then $A_1$ is similar to $A_3$. To see that, write $A_2=P^{-1}A_1P$ and $A_3=Q^{-1}A_2Q$. Then $A_3=Q^{-1}P^{-1}A_1PQ=(PQ)^{-1}A_1PQ$ and $PQ$ is invertible.
Now, to solve the problem, note that $P^{-1}AP$ is similar to $A$ and $Q^{-1}BQ$ is similar to $B$.
if $A \sim B$ then there exists an invertible $R$ such that $A=R^{-1}BR$ hence given any invertible matrices $P$ and $Q$ we have that $P^{-1}AP=(P^{-1}R^{-1}Q)Q^{-1}BQ(Q^{-1}RP)$ which implies that $P^{-1}AP\sim Q^{-1}BQ$