TITLE: Must a matrix of which all conjugates have zero diagonal be zero?
QUESTION [16 upvotes]: Let $A$ be an $n \times n$ real matrix with the following property:
All the conjugates of $A$ have only zeros on the diagonal. Does $A=0$?
(By conjugates, I mean all the matrices similar to it, over $\mathbb{R}$, that is I require the conjugating matrix to be real).

Of course, if $A$ is diagonalizable, then clearly it must be zero.
The only idea I have is to use the Jordan form for real matrices, but after some thought I am not sure this is a good approach.

REPLY [5 votes]: The field is unimportant. The given condition implies that $\operatorname{tr}(APDP^{-1}) = \operatorname{tr}(P^{-1}APD) = 0$ for every invertible matrix $P$ and every diagonal matrix $D$. Since the set of all diagonalisable matrices spans the whole matrix space (there is actually an even better result, namely, every square matrix is the sum of at most three diagonalisable matrices), the previous observation in turn implies that $\operatorname{tr}(AB)=0$ for every matrix $B$. Hence $A=0$.