Proving that $\chi_{T^*}=\overline{\chi_T}$ and $m_{T^*}=\overline{m_T}$ (characteristic and minimal polynomials of adjoint map)?

Question

For a linear map $T:V\to V$ where $V$ is a finite dimensional inner product space over $\mathbb{C}$, I know the result $\chi_{T^*}=\overline{\chi_T}$ (where $T^*$ is the adjoint map for $T$). My problem is I can't find a proof for it... Do you know one?

Also, I'm guessing we can also prove $m_{T^*}=\overline{m_T}$, but again, can't find a proof anywhere or generate one myself...

Can you help me please? I really don't see...

(here $\chi$ is for characteristic polynomial and $m$ for the minimal one)

Answer 1

The minimal and characteristic polynomials of (a square matrix) $A$ are both invariant under transposition of$~A$. For the characteristic polynomial this is because the determinant is invariant, and for the minimal polynomial because clearly $P[A^T]=P[A]^T$ for every (complex) polynomial $P$.

So instead of $T^*$ you can work with its transpose $\overline T$. But $\chi_{\overline T}=\overline{\chi_T}$ and $m_{\overline T}=\overline{m_T}$ are obvious because complex conjugation is a ring automorphism of$~\Bbb C$.

Answer 2

Hint: What is the relationship between the eigenvalues of $T$ and $T^*$?