I have a 10x10 symmetrical 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, 5x5 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.

Thanks,

Fiona

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.