# Correct way to calculate a matrix trace with negative values

I have a $$10 \times 10$$ symmetric variance-covariance matrix, such that the variances for $$10$$ vectors are on the main diagonal and the covariance between all vectors are on the off-diagonals.

I want to quantify the amount of variance in total. I can easily take the matrix trace as the sum of the eigenvalues on the main diagonal.

However, the matrix can be split into meaningful (biologically meaningful, in my case) sub-matrices: $$4$$ submatrices, $$5 \times 5$$ each, in each corner of the original matrix. If I then want to quantify the variation within each sub-matrix using the matrix trace, I run into some trouble with the top-right/bottom-left sub-matrices. These are formed of covariance estimates and are therefore not necessarily positive. My question is, what is the correct way to calculate the matrix trace here? If I sum the eigenvalues, I will have some negative values subtracting from the total, so should I use absolute values? Is the matrix trace the best method to use here or is there a more appropriate way of summarising the amount of variance in the sub-matrices?

Any guidance would be gratefully received.

• Something seems off here. If you are working with a covariance matrix it should be semi-positive definite. One property a matrix like this must have is the eigenvalues must be non-negative. Oct 1, 2022 at 3:52

If you want something eigenvalues driven choose (assumed $A$ is real valued) $$\|A\|_2=\sqrt{\max_n{\lambda_n(AA^T)}}$$ where $\lambda_n(AA^T)$ denotes the n'th eigenvalue of matrix $AA^T$.