I am generating correlation matrix by some new algorithm. Generated matrix is non positive semi-definite matrix.
I’m getting a few negative eigenvalues. The rest of eigenvalues are quite equal to the ideal matrix.
Can I use that non positive semi-definite matrix? If not, why?
If my estimated correlation matrix has all positive but complex value and imaginary terms are close to zero then is it possible?
A correlation matrix is really the covariance matrix of a bunch of variables which have been rescaled to have variance one.
But every population covariance matrix is positive semi-definite, and if we rule out weird cases (such as with missing data, or “numerical fuzz” turning a small eigenvalue to a negative one), so is every sample covariance matrix.
So if a matrix is supposed to be a correlation matrix, it should be positive semi-definite.
Note that the semi-definite is important here. In the bivariate case, take your two variables to be perfectly positively correlated and then the correlation matrix is (1111) which has eigenvalues of 2 and 0: the zero eigenvalue means it is not positive definite.