However due to inexactness of floating point numerical computations, even algebraically positive definite cases might occasionally be computed to not be even positive semi-definite good choice of algorithms can help with this. The same should generally apply to covariance matrices of complete samples (no missing values), since they can also be seen as a form of discrete population covariance. Population covariance matrices are positive semi-definite. Their covariance matrix, $M$, is not positive definite, since there's a vector $z$ ($= (1, 1, -1)'$) for which $z'Mz$ is not positive. Consider three variables, $X$, $Y$ and $Z = X+Y$.
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