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But 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.

Sample covariance matrices - depending on how they deal with missing values in some variables - may or may not be positive semi-definite. If pairwise deletion is used, for example, then there's no guarantee of positive semi-definiteness. Further, accumulated numerical error can cause sample covariance matrices that should be notionally positive semi-definite to fail to be.

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.

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.

More generally, sample covariance matrices - depending on how they deal with missing values in some variables - may or may not be positive semi-definite, even in theory. If pairwise deletion is used, for example, then there's no guarantee of positive semi-definiteness. Further, accumulated numerical error can cause sample covariance matrices that should be notionally positive semi-definite to fail to be.

This happened on the first example I tried (I probably should supply a seed but it's not so rare that you should have to try a lot of examples before you get one).

Sample covariance matrices - depending on how they deal with missing values in some variables - may or may not be positive semi-definite. If pairwise deletion is used, for example, then there's no guarantee of positive semi-definiteness. Further, accumulated numerical error can cause sample covariance matrices that should be notionally positive semi-definite to fail to be.

But 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.

Sample covariance matrices - depending on how they deal with missing values in some variables - may or may not be positive semi-definite. If pairwise deletion is used, for example, then there's no guarantee of positive semi-definiteness. Further, accumulated numerical error can cause sample covariance matrices that should be notionally positive semi-definite to fail to be.

 z <- x + y + rnorm(30)/50  # same x and y as before.
 xyz1 <- data.frame(x=x,y=y,z=z) # high correlation but definitely of full rank
 xyz1$x[sample(1:30,5)]<-NA   # make 5 x's missing
 xyz1$y[sample(1:30,5)]<-NA   # make 5 y's missing
  xyz1$z[sample(1:30,5)]<-NA   # make 5 z's missing
  cov(xyz1,use="pairwise")     # the individual pairwise correlations are fine ...
           x          y        z
x  1.2107760 -0.2552947 1.255868
y -0.2552947  1.2728156 1.037446
z  1.2558683  1.0374456 2.367978
 chol(cov(xyz1,use="pairwise"))  # ... but leave the matrix not positive semi-definite
Error in chol.default(cov(xyz1, use = "pairwise")) : 
  the leading minor of order 3 is not positive definite

 chol(cov(xyz1,use="complete")) # but deleting even more rows leaves it PSD
          x          y          z
x 0.8760209 -0.2253484 0.64303448
y 0.0000000  1.1088741 1.11270078
z 0.0000000  0.0000000 0.01345364

z <- x + y + rnorm(30)/50  # same x and y as before.
xyz1 <- data.frame(x=x,y=y,z=z) # high correlation but definitely of full rank 

xyz1$x[sample(1:30,5)] <- NA   # make 5 x's missing  

xyz1$y[sample(1:30,5)] <- NA   # make 5 y's missing   

xyz1$z[sample(1:30,5)] <- NA   # make 5 z's missing   

cov(xyz1,use="pairwise")     # the individual pairwise covars are fine ... 

           x          y        z
x  1.2107760 -0.2552947 1.255868
y -0.2552947  1.2728156 1.037446
z  1.2558683  1.0374456 2.367978 

 chol(cov(xyz1,use="pairwise"))  # ... but leave the matrix not positive semi-definite 

Error in chol.default(cov(xyz1, use = "pairwise")) : 
  the leading minor of order 3 is not positive definite

 chol(cov(xyz1,use="complete")) # but deleting even more rows leaves it PSD 

          x          y          z
x 0.8760209 -0.2253484 0.64303448
y 0.0000000  1.1088741 1.11270078
z 0.0000000  0.0000000 0.01345364