Definition of a Multivariate Normal random variable

Question

According to Wikipedia, $$ {\displaystyle \mathbf {X} \ \sim \ {\mathcal {N}}(\mathbf {\mu } ,{\boldsymbol {\Sigma }}) \iff {\text{there exist }}\mathbf {\mu } \in \mathbb {R} ^{k},{\boldsymbol {A}}\in \mathbb {R} ^{k\times \ell }{\text{ such that }}\mathbf {X} ={\boldsymbol {A}}\mathbf {Z} +\mathbf {\mu } {\text{ for }}Z_{n}\sim \ {\mathcal {N}}(0,1),{\text{ i.i.d.}}} $$ For a non-degenerate distribution, does $k$ always have to be equal to $l$?
Some texts like this and this use $k = l$ directly in the definition itself, but I can't see why it's necessary

Answer 1

If we want $\mathbf{X}$ to be non-degenerate, we should check whether the rank of the $k\times k$ matrix $Cov(\mathbf{X})=\mathbf{A}\mathbf{A}^{\top}$ is $k$.

Since $rank(\mathbf{A}\mathbf{A}^{\top})=rank(\mathbf{A})\le \min (k,l)$, $\mathbf{X}$ must be degenerate when $k>l$.

We can find an example to show $\mathbf{X}$ could be non-degenerate when $k<l$: just take $k=1$, $l=2$ and $\mathbf{A}=\begin{bmatrix}1\\0\end{bmatrix}$.