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Let ##A \in \mathbb C^{m\times n}##. Prove that tr##(A^*A)=0## if and only if ##A^*A=0## (here ##0## obviously means the zero matrix).

The attempt at a solution.

By definition of the trace of a matrix, the implication ← is obvious. I am having problems proving the other implication: first of all, I have doubts about the meaning of ##A^*##, if ##f:\mathbb C^n \to \mathbb C^m## is a linear transformation and ##A## is the associated matrix to ##f##, then ##f^t:{(\mathbb C^m)}^* \to {(\mathbb C^n)}^*## defined as ##f^t(\phi)=\phi \circ f## for all ##\phi \in {(\mathbb C^m)}^*##, so, I suppose ##A^*## just means the matrix which represents ##f^t##, am I right?

I have no idea how to prove ##tr(A^*A)=0 \implies A^*A=0##. I would appreciate any suggestions.