What's an intuitive way to think about the determinant? In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a $2\times 2$ matrix by the formula. Our teacher showed us how to compute the determinant of an $n \times n$ matrix by breaking it up into the determinants of smaller matrices. Apparently there is a way by summing over a bunch of permutations. But the notation is really hard for me and I don’t really know what’s going on with them anymore. Can someone help me figure out what a determinant is, intuitively, and how all those definitions of it are related?

If you have a matrix 


*

*$H$
then you can calculate the correlationmatrix with 

*$G = H \times H^H$
(H^H denotes the complex conjugated and transposed version of $H$).


If you do a eigenvalue decomposition of $G$ you get eigenvalues $\lambda$ and eigenvectors $v$, that in combination $\lambda\times v$ describes the same space.
Now there is the following equation, saying:


*

*Determinant($H*H^H$) = Product of all eigenvalues $\lambda$


I.e., if you have a $3\times3$ matrix $H$ then $G$ is $3\times3$ too giving us three eigenvalues.
The product of these eigenvalues give as the volume of a cuboid.
With every extra dimension/eigenvalue the cuboid gets an extra dimension.
