Let $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?

Solution 1:

This is true if the underlying field is of characteristic zero. Over such a field, every traceless matrix $A$ is similar to a matrix $B$ with a zero diagonal (see Kahan's proof, for instance). Now, let $A=PBP^{-1}$. Split $B$ into the sum of its strictly upper triangular part $U$ and its strictly lower triangular part $L$. Then $A=PUP^{-1}+PLP^{-1}$ is a desired split.

The statement is not necessarily true over a finite field. For a counterexample, consider $I_2$ over $F=GF(2)$. The set of all nilpotent matrices in $M_2(F)$ is $$ S=\left\{0,\,\pmatrix{0&1\\ 0&0},\,\pmatrix{0&0\\ 1&0},\,\pmatrix{1&1\\ 1&1}\right\}. $$ No two matrices in $S$ add up to $I_2$.