Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$. Show that an upper triangular matrix in $\mathcal{M}_n(\mathbb{C})$ with zero diagonal is nilpotent and that a nilpotent matrix is similar to an upper triangular matrix with zero diagonal.
Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Show that an upper triangular matrix in $\mathcal{M}_n(\mathbb{C})$ with zero diagonal is nilpotent and that a nilpotent matrix is similar to an upper triangular matrix with zero diagonal.