This is the standard, elementary arithmetic proof that the marginal and conditional distributions of the multivariate Gaussian are again Gaussian with parameters expressible in terms of the covariance matrix of the original Gaussian. We use the block multiplication of matrices.

I was surprised by how much work is required to show this, and feel moreover that the proof (while correct) fails to offer any intuitive understanding. Is there not a higher-level, co-ordinate free proof of this important result, perhaps one that uses characteristic properties of multivariate Gaussians?

**Update:** I received an answer to this question on Cross Validated from whuber. He uses a generative definition: the multivariate Gaussians are precisely the affine transformations of tuples of standard (mean zero, unit-variance) one-dimensional Gaussians. Using this definition, it follows quickly that the conditional and marginal distributions must be multivariate Gaussian.