Solved – Is every correlation matrix positive semi-definite

correlation matrixcovariance-matrixlinear algebra

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?

Best Answer

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 $\pmatrix{1 & 1 \\ 1& 1}$ which has eigenvalues of $2$ and $0$: the zero eigenvalue means it is not positive definite.