This is from Hoffman and Kunze 6.4.13. I am studying for an exam and trying to solve some problems in Hoffman and Kunze.
Here is the question.
Let $V$ be the space of $n\times n$ matrices over a field $F$. Let $A$ be a fixed matrix in $V$. Let $T$ and $U$ be linear operators on $V$ defined by
$T(B)=AB$ and $U(B)=AB-BA$.
a) (True or False) If $A$ is diagonalizable then $T$ is diagonalizable.
This is true. I can show that both A and T have the same minimal polynomial. So if
$A$ is diagonalizable then the minimal polynomial of $T$ should be a product of distinct linear factors. Proving that $T$ is diagonalizable. but its the next question I am having trouble with.
b)(True or false) If $A$ is diagonalizable then $U$ is diagonlizable.
I am thinking this is false. But I can't think of of a counter example. May be I am wrong.
Can anybody help?. Thanks for all your help.
Best Answer
(a) True. Indeed $p(T)B = p(A)B$ for every polynomial in $F[X]$ whence $p(T) = 0$ if and only if $p(A)=0$. By the minimal polynomial characterization of diagonalizability in finite dimension (splits with simple roots), it follows that $T$ is diagonalizable if and only $A$ is diagonalizable.
(b) True. If $A=P^{-1}DP$ is diagonalizable, then $U$ is similar to $V(B)=DB-BD$ with $D=\mbox{diag}(d_1,\ldots,d_n)$ diagonal. Precisely, $U=S\circ V\circ S^{-1}$ with $S(B)=P^{-1}BP$.
Then denoting $E_{ij}$ the canonical basis of $M_n(F)$, we find $$V(E_{ij})=DE_{ij}-E_{ij}D=(d_i-d_j)E_{ij}.$$ Whence $V$ is diagonal in the canonical basis, and $U$ is therefore diagonalizable.