For an arbitrary complex matrix A show that $$A*A^\dagger$$ is Hermitian.
Where the dagger "$\dagger$" stands for the "complex conjugate and transpose" operators.
From what I understand this must mean that $$A*A^\dagger = [A*A^\dagger]^\dagger$$ But I am stuck. I don't really understand the properties of the complex conjugate function with Matrices.
Best Answer
The key properties you need are the fact that hermitian-conjugating twice returns you to where you started, $$(A^\ast)^\ast=A$$ (which holds because it holds for both transposing and complex-conjugating), and the formula for the conjugate of a product, $$(AB)^\ast=B^\ast A^\ast,$$ which is inherited from the identical transpose formula and is unchanged by complex conjugation.
Of course, you can also prove both formulas from the (real) definition of the conjugate of $A$, namely that for all $u$ and $v$ you get $\langle A u,v\rangle=\langle u, A^\ast v\rangle$, as Nick points out.