Let $(V, \mathbb{C})$ be a complex - valued vector space. Let $A$ be any linear operator acting on this vector space. Suppose that $B = \{|v\rangle_{k}\}_{k=1}^{n}$ is a basis set for $(V, \mathbb{C})$. Then, the linear operator $A$ may be written as $A = \sum_{k=1}^{n}A_{kj}|v_{k}\rangle \langle v_{j}|$ - in terms of it's outer product.
I would appreciate hints in the direction of assisting me in writing $A = B + iC$ where $B,C$ are Hermitian and $i$ is an imaginary number.
One can write $A$ as the sum of a real symmetric matrix, a pure imaginary symmetric matric, a real antisymmetric matrix, and a pure imaginary antisymmetric matrix. Each of these 4 kinds of pieces can be written in the form $B+iC$ for Hermetian $B$ and $C$, hence so can $A$.