Semi-definite Covariance Matrix For distributions

Hi all, recently I am dealing with state space model. For the transition equation x_t = Ax_{t-1} + W w_t, the w_t~N(0, I) is a multivariate normal distribution and W is a (possibly drop-rank) noise matrix. So I want to use MultivariateNormalLinearOperator to generate the a distribution object. For example, the state dimension is 6, W is 6x3 matrix, therefore, MultivariateNormalLinearOperator(loc=tf.zeros([6,]), scale=W) would raise Error when using .sample() because of incorrect dimension.

So I change to use MultivariateNormalFullCovariance with covariance matrix W @ tf.transpose(W). However, the covariance matrix not a full rank matrix, thus positive semi-definite. The nan samples happen when using .sample(). Is there any good method to solve this problem?

Start by reproducing the reported .sample() behavior with MultivariateNormalLinearOperator, a 6-dimensional location, and a [6, 3] matrix W. Compare it with MultivariateNormalFullCovariance using W @ tf.transpose(W) and determine the expected handling of a positive semi-definite covariance; done should be a clear supported behavior or documented limitation.