What if the covariance matrix is singular?

[Yu-Xiaoxian]

In Ch.3 assignment 1.3， at t=1, the covariance matrix [0.25, 0.5; 0.5, 1.0] is a singular matrix. When we want to calculate its gaussian probabilistic, we need the matrix to be invertible. How can we deal with this situations.

[pptacher]

Hi sorry for slow response. In 1 dimension when σ → 0 the normal distribution with mean 0 converges (in distribution, weak convergence) to the delta dirac distribution: all the probability mass is concentrated in 0.
We can generalize in 2d when 1 eigen value of the covariance matrix is null the 2d gaussian actually degenerates to a 1d gaussian with all probability mass concentrated on a line (direction defined by eigenvector corresponding to the non null eigenvalue).(corrected a typo in the P matrix). So the uncertainty ellipse becomes a bounded interval on a line.
Does this clarify ? is it the point you wanted to make ?

[Yu-Xiaoxian]

Your replay is beyond my point. The probability mass center explain the covariance matrix very clear, thanks.

When I use random module of Python to sample from normal distribution, I add a tiny diagonal matrix to the covariance matrix, making it ill but not singular, so it becomes invertible. And the result shows exactly what you've said, the sample points centered in a line.

Thanks again for your replay, that makes the result more clear.