There's a geometric interpretation of the determinant of an operator in a real vector space that I've always found intuitive. Suppose we have a n-dimensional real-valued vector space. We can plot n vectors in an n-dimensional Cartesian coordinate system, and in general we'll have an n-dimensional parallelepiped. If we apply an operator to these n vectors, the result is a new parallelepiped whose "n-dimensional volume" is equal to the volume of the original parallelepiped time the (absolute value of the) determinant of the operator. Is there an extension of this interpretation for an operator in a complex vector space?