# What's the geometric intuition of the determinant of a covariance matrix?

As explained in intuition of determinant : we know that determinant is the how the volume is scaled when the matrix is regarded as a projection.

Now, if we narrow down the definition of the matrix here: assuming that it is a covariance matrix, i.e., positive semi definitive and symmetric, will there be any new characteristics/intuition of determinant.

If $$\rho_{ij}=\operatorname{Cov}(X_i,\,X_j)$$ and $$Y_i:=R_{ij}X_j$$ for a rotation matrix $$R$$ then $$\operatorname{Cov}(Y_i,\,Y_j)=R_{ik}\rho_{kl}R^T_{lj}$$, which for a suitable choice of $$R$$ is diagonal. So $$\det\rho$$ is the determinant of this diagonal matrix, i.e. the product of variances of the $$Y_i$$. This rotation places Cartesian axes along the axes of the ellipse associated with $$\rho$$, so the rotation-invariant determinant, being a product of eigenvalues, is proportional to the ellipse's area. (Needless to say, in higher dimensions I'd be talking about a hyperellipsoid's measure, hypervolume, volume, or whatever term you prefer.)