# 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?

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.