Suppose we map the low dimensional Gaussian distribution into higher dimension using linear transform. Say, $X \in R^p$ is joint Gaussian, and for $n > p$, $Y = A_{n \times p}X$. Is $Y$ joint Gaussian?
I know if we map $X$ into equal or lower dimension using linear transform, the new variables are joint Gaussian. My question is what if we map $X$ into higher dimension?
My professor told me it's Gaussian, and it's degenerate Gaussian. But is it always Gaussian or degenerate Gaussian? WHO can give me analytical proof!
What makes the difference here is that the matrix involved is not singular any more. So this case is much harder.
I hope I'm clear:P
If you map a vector space into a vector space of higher dimension, the image is contained in proper subspace of dimension at most the dimension of the original vector space. Thus mapping into a higher dimension is equivalent to mapping into a vector space of equal or lower dimension.