let $\mathbb{H}$ be the quaternions and $Mat(n,\mathbb{H})$ be the vector space of $n\times n$ matrices over $\mathbb{H}$. Let $H(n,\mathbb{H}):=\{ A\in Mat(n,\mathbb{H}): \overline{A}^t=A \}$ be the subspace of all hermitian matrices over $\mathbb{H}$.
1)My first question: the hermitian matrices are no vector space over $\mathbb{H}$, since if A is a hermitian matrix, so $i\cdot A$ is not hermitian anymore. What does the author really mean by this.
(...) 2)Later on, the author says, that if $A,B\in H(n,\mathbb{H})\Rightarrow Tr(AB^*)$ is real. But this is also not clear for me.
We have $Tr(AB^*)=Tr(AB)$ but if we consider two matrices like:
\begin{pmatrix} 1 & i\\ -i & 1\end{pmatrix} and \begin{pmatrix} 1 & j\\ -j & 1\end{pmatrix}
Then we see, that the Trace is not real. What's wrong in my understanding?
I hope, you can help me.
For (1): A Hermitian matrix $A$ here must have a real diagonal, and the diagonal entries of $iA$ are not real.
For (2), it looks like you have a counterexample for the statement as written. It's a little strange that the $^*$ would appear if the statement were about Hermitian matrices... why write it at all?
It's true though that $tr(AA^*)$ is real for any $A$