# Geometrical intuition behind complex determinant [closed]

I'm hoping to gain some insight into the geometric meaning of the determinant of a complex square matrix. Is there anything analogous to the intuition behind the determinant of a real matrix as the volume of the image of a unit cube?

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I'm not sure I understand why this was closed. There are many questions asking for intuition on MO, such as mathoverflow.net/questions/13526/…. Is this due to a change in policy since similar questions were posted? –  Benjamin Jan 14 at 22:21
I think the perception is that the geometric meaning of the determinant is well-understood (indeed one would hope that whoever introduces the determinant in an undergraduate course would take time to mention the geometry), whereas the question you link to is about the trace whose geometric meaning is a little more subtle. In other words your question is not at research-level - does that clarify things? –  Nick Gill Jan 16 at 10:23
To be honest, I received no geometric interpretation for the complex determinant as an undergraduate and the best I could find online (after only a quick search) was math.stackexchange.com/q/140314/103279. –  Benjamin Jan 21 at 14:32
This answer talks about the determinant in a way you might find useful: mathoverflow.net/questions/7584/… –  Nick Gill Jan 21 at 15:01
I'm still not really seeing anywhere that discusses the sense in which the complex determinant specifically is a volume? In what notion of volume? I was hoping for an answer similar to the ones here mathoverflow.net/q/79907/41654. In what sense exactly is the complex the determinant a volume, for what sense of volume? How does this volume in \mathbb{C}^n relate to volumes in $\mathbb{R}^{2n}$? Are they the volumes coming from the standard inner products of both spaces? –  Benjamin Jan 21 at 15:13