The determinant of a real-valued matrix can be seen as the volume of the parallelotope with the column vectors as the sides. Is there an analogous interpretation for complex-valued matrix determinants? I'm not entirely sure what a complex-valued vector would look like and thus how to build a parallelotope out of one, but maybe there's some other interpretation.
Looking at some similar questions, I'm wondering if Clifford algebra could be the key here? I'm not sure. I don't think I've worked with complex spaces enough to be able to figure it out on my own.
Any thoughts?
Ok if nobody else is going to address this one let me give it a try.
In complex numbers the real and imaginary components operate by different rules. Multiplication by the real component scales a vector in or out from the origin, whereas multiplication by the imaginary component rotates it about the origin.
In Clifford Algebra there is no distinction between real and imaginary components, all dimensions work by the same rules. Multiplying parallel vectors scales their length in or out from the origin, whereas multiplication of non-parallel vectors results in a bivector product with a "twist" that rotates through the angle from the first vector to the second.
The volume of the parallelotope due to multiplication of the row-vectors uses "real" multiplication, i.e. without an imaginary component. The Wiki page on the determinant says
"The bivector magnitude (denoted (a, b) ∧ (c, d)) is the signed area, which is also the determinant ad − bc."
http://en.wikipedia.org/wiki/Determinant
I presume the answer to your question about a "complex" determinant would be the bivector itself, not just the "bivector magnitude", i.e. not just the area of the parallelogram in the 2-D example, but also its "twist", the rotational component of the multiplication.
As to what that actually "means" is open to interpretation. The utility of the determinant is what it tells you about the matrix, or how it "represents the image of the unit square under the mapping" (from the Wiki page). So I presume the "complex determinant" tells you not only how the matrix would scale the unit square, but how it would twist it aswell.