The question is as follows, where A is an n by n matrix:
Show that if A has all real eigenvalues, then A = B + C, where B is symmetric and C is nilpotent.
I have tried thinking about this in terms of the triangulation theorem, but I cannot get this proof started. How do I show this?
Use the Schur decomposition. If $A$ is real with real eigenvalues, then there exists an orthogonal matrix $U$ and upper triangular $M$ such that $A = UMU^T$. Now, take $B = UM_1U^T$ and $C = UM_2U^T$ where $$ M_1 = \pmatrix{m_{11}\\ & \ddots \\&&m_{nn}}, \quad M_2 = \pmatrix{0& m_{12} & \cdots & m_{1n}\\ &0 & \ddots & \vdots\\ &&\ddots& m_{(n-1)\,n}\\ &&&0}. $$