Paper 1, Section II, 7B

(a) Let $A$ be an $n \times n$ matrix. Define the characteristic polynomial $\chi_{A}(z)$ of $A$. [Choose a sign convention such that the coefficient of $z^{n}$ in the polynomial is equal to $\left.(-1)^{n} .\right]$ State and justify the relation between the characteristic polynomial and the eigenvalues of $A$. Why does $A$ have at least one eigenvalue?

(b) Assume that $A$ has $n$ distinct eigenvalues. Show that $\chi_{A}(A)=0$. [Each term $c_{r} z^{r}$ in $\chi_{A}(z)$ corresponds to a term $c_{r} A^{r}$ in $\left.\chi_{A}(A) .\right]$

(c) For a general $n \times n$ matrix $B$ and integer $m \geqslant 1$, show that $\chi_{B^{m}}\left(z^{m}\right)=\prod_{l=1}^{m} \chi_{B}\left(\omega_{l} z\right)$, where $\omega_{l}=e^{2 \pi i l / m},(l=1, \ldots, m) .[$ Hint: You may find it helpful to note the factorization of $z^{m}-1$.]

Prove that if $B^{m}$ has an eigenvalue $\lambda$ then $B$ has an eigenvalue $\mu$ where $\mu^{m}=\lambda$.

*Typos? Please submit corrections to this page on GitHub.*