So I was working on a specific problem related to Hermitian matrices. If we let $H_n$ denote the set of n x n Hermitian matrices. We're told that $H_n$ is a real vector space under matrix addition and scalar multiplication by a real number. I don't understand why though, because what would prevent you from adding two imaginary numbers?
Then we're told to write a basis for $H_2$. Now for a Hermitian matrix, it's equal to its conjugate transpose, so a basis would be:
$\left(\begin{array}{ccc} 0 & 1 \\ 1 & 0\\ \end{array}\right) , \left(\begin{array}{ccc} 0 & 0 \\ 0 & 1\\ \end{array}\right), \left(\begin{array}{ccc} 1 & 0 \\ 0 & 0\\ \end{array}\right), \left(\begin{array}{ccc} 0 & i \\ -i & 0\\ \end{array}\right)$
Which should be the same as the basis for a symmetric matrix, correct?
From there though, we have $sH_n$ represent skew-Hermitian matrices. I believe the basis for $sH_n$ is:
Edit: updated my basis here.
$\left(\begin{array}{ccc} 0 & -1 \\ 1 & 0\\ \end{array}\right), \left(\begin{array}{ccc} 0 & i \\ i & 0\\ \end{array}\right)$
because the diagonals would have to be zero right, to make the conjugate transpose of A be equal to -A?
From there though, they ask if $sH_n$ is a real vector space, and I'm not sure what to answer. It's real if we only use reals?
I'm asked the same thing for $U_n$, where $U_n$ represents n x n unitary matrices. I'm not even sure how to come up with a basis for this set??
Edit: Technically, the question says "Is $U_n$ a real vector space? Write a basis if possible."
I suppose I couldn't write a basis because $U_n$ is not a real vector space then. How would I show that though?
Thanks in advance for your help.