TITLE: Does it make sense to talk about complex matrices over the field of real numbers, R?
QUESTION [5 upvotes]: I don't see an issue with considering a vector space of complex matrices over R -- addition of matrices makes sense, but scalar multiplication will be done with real numbers.
But I wanted to ask, just in case I may have it wrong.
Thanks,
Edit: As far as I know, the entries being complex does not make it a vector space over C - it is the numbers that we are using to scale the matrices that determine the ground field.  (But like I said, I may have it wrong...)

REPLY [2 votes]: The vector space is actually a $4$-uple $(V, \Bbb F, +, \cdot)$, where $V$ is a set, $\Bbb F$ is the underlying field, and $+$ and $\cdot$ are operations $+: V\times V \to V$, $\cdot : \Bbb F \times V \to V $ satisfying some conditions. We abuse notation and call $(V,\Bbb F, +, \cdot)$ only $V$, the rest being understood from context.
If you have a complex vector space, that is, $(V, \Bbb C, +,\cdot)$, then $(V, \Bbb R, +, \cdot\big|_{\Bbb R \times V})$ is a real vector space. Note that $V$ and $+$ are the same in both cases. That's all there is to it.