We know that a square matrix is a covariance matrix of some random vector if and only if it is symmetric and positive semi-definite (see Covariance matrix). We also know that every symmetric positive definite matrix is invertible (see Positive definite). It seems that the inverse of a covariance matrix sometimes does not exist.
Does the inverse of a covariance matrix exist if and only if the covariance matrix is positive definite? How can I intuitively understand the situation when the inverse of a covariance matrix does not exist (does it mean that some of the random variables of the random vector are equal to a constant almost surely)?
Any help will be much appreciated!