I'm aware that positive definite matrix is invertible. I've also read that a matrix is semi-positive definite if and only if the matrix is invertible.
Then, is my matrix invertible when it is not positive definite, nor not semi-positive definite?
One more question - I've encountered a situation in which I have non-positive definite matrix due to floating point precision issues: some of my elements were $-1.623$e-16.
I am computing a covariance matrix, so I know that my matrix should theoretically be semi-positive definite. Therefore, $-1.623$e-16 is negative due to rounding errors. In a situation like this, is it safe to replace it with $0$?