Page "Multivariate normal distribution" Paragraph 104

Conversely, any choice of μ, full rank matrix U, and positive diagonal entries Λ < sub > i </ sub > yields a non-singular multivariate normal distribution.

If any Λ < sub > i </ sub > is zero and U is square, the resulting covariance matrix UΛU < sup > T </ sup > is singular.

Geometrically this means that every contour ellipsoid is infinitely thin and has zero volume in n-dimensional space, as at least one of the principal axes has length of zero.