I have a covariance matrix (S) which was computed using
where C is a diagonal matrix and J is another matrix. Both C and J are functions of some parameters. My problem is that the matrix S should be symmetric as is clear from the above expression. But MatLab does not return a symmetric covariance matrix. I checked it by computing Q:
Now, all the elements of Q should be zeros. But they are not as you can see in attached Q.mat file. I understand that this may be due to small some floating point arithmetic but I need it to be exactly symmetric because otherwise, my eigenvalues become complex which is physical implausibility. I have tried a fix for this
It makes my matrix symmetric. But the next problem is that since it is a covariance matrix it should be positive-definite (at-least semi-positive definite) but it gives me very small negative eigenvalues which again might be due to floating point arithmetic. So I fix this again by forcefully making the negative eigenvalues equal to zero as follows
It removes the negative eigenvalue problem but it again makes the matrix not exactly-symmetric. So it seems that I am trapped in this cycle. Any suggestion for possible solutions?