I am having some trouble understanding a definition/question in my linear algebra text book.
The question states " If $A$ is a square matrix, a matrix of the form $P^{-1}AP$ where $P$ is invertible is called a 'Conjugate ' of $A$.
I am having trouble understanding that, is there any examples? It then asks is $P^{-1}AP=A$ and I'm not sure what direction to take. I tried $P(P^{-1}AP=A)$ , giving $AP=PA$ and then $(AP=PA)P^{-1}$ giving $A=PAP^{-1}$. But I'm not sure if this is a correct proof/ not sure what I am showing. It also asks if some conjugate of $A$ is invertible show that $A$ is invertible, not sure where to begin on that either.
Thanks so much for any help
Best Answer
1) $A$ is conjugate to itself, simply choose $P = I$ the identity matrix.
2) Suppose some conjugate of $A$ is invertible $\implies \det(P^{-1}AP) \neq 0 \implies \det(A) \neq 0$ which shows that $A$ is also invertible.