Distance of a covariance matrix from a perfect 1:1 relationship

Question

I have a number of estimated variance-covariance matrices, and I would like to know how different these are from a perfect 1:1 relationship. To be specific:

• They are (genetic) variance-covariances for males and females for the same trait.
• By perfect 1:1 relationship I mean that the correlation is 1, and that the variances are equal. So that all data would fall along y = x.
• I do not care about the scale of the variance-covariance matrix. The ellipse can be smaller or larger, but I do care about differences in variance between the two axes.

The context is that I'd like to know if you'd move e.g. 1 standard deviation in this 2D space, how much you expect to move away from equality between the axes (y = x).

I am imagining that if you would draw the variance-covariance matrix as an ellipse centered on (0, 0), there would be some sort of average distance to the line y = x. I suppose I could take a 100 points along the ellipse and calculate that distance, but wondering is there is something more straightforward available.

Does anyone know an existing distance metric that would make sense? Or another solution?

Answer 1

For covariance matrices there is a distance metric called Riemannian distance. It's the length of geodesic in the space of positive definite matrices. The geodesic preserves the determinant of the matrix. The definition in Encyclopedia of Distances Section 12.3 is as follows: $$Sym^+(A,B)=\left(\sum_i\log^2\lambda_i\right)^{\frac 1 2},$$ where $\lambda_i$ - eigenvalues of matrix $AB^{-1}$.

For correlation matrices I don't find this metric applicable because they are on their own manifold within symmetric positive definite matrices space where covariance matrices live. So, for instance on this manifold the trace should be constant, not just the determinant. There is a metric of correlation matrices too. It has been suggested in his PhD thesis "A Riemannian Quotient Structure for Correlation Matrices with Applications to Data Science" by Paul David.