Show that if $F(x)$ is a polynomial over $K$, then $F(A)=0$ if and only if $F(T_A)=0$.

Question

Let $V$ be the space of $n\times n$ matrices over a field $K$. Let $A$ be an $n\times n$ matrix. Let $T_A: V\to V$ be the linear transformation given by $T_A(B)=AB$.

Show that if $F(x)$ is a polynomial over $K$, then $F(A)=0$ if and only if $F(T_A)=0$.

---

We can easily show that if $F(T_A)=0$, then $F(A)=0$ since we can consider $F(A)=F(T_A(I))=0$.

---

However, the converse I don't think is correct. Let's say we are over $\mathbb R$.

If we have the matrix $A=\begin{pmatrix}0&2\\2&0\end{pmatrix}$, and the polynomial $F(x)=(x-2)(x+2)(x+3)$, then $F(A)=0$ since $\mu_A(x) \mid F(x)$, but if we let $B=\begin{pmatrix}2&3\\1&3\end{pmatrix}$, then $F(AB)\ne 0$. So, it cannot be the case that $F(T_A)=0$.

Is this correct or am I missing something here?

Answer 1

Remark that $X^n(T_A)(B)=(T_A)^n(B)=A^nB$, thus $P(T_A)(B)=P(A)B$.