# Normal n-distribution and Variance-covariance matrix. Let x be a standard normal vector of length n. Let A in R^{m times n} be a matrix.

Normal n-distribution and Variance-covariance matrix
Let x be a standard normal vector of length n.<br.Let $A\in {\mathbb{R}}^{m×n}$ be a matrix.
Then we say that Ax has normal distributed components. Or we could say that Ax is a vector whose components are normally distributed with length m.
Example:
$A=\left[\begin{array}{cc}3& -2\\ 2& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}x{=}^{\left(d\right)}N\left(O,{I}_{2×2}\right)=\left[\begin{array}{c}x\\ y\end{array}\right]\phantom{\rule{0ex}{0ex}}Ax=\left[\begin{array}{c}3X-2Y\\ 2X+Y\end{array}\right]$
$3X-2Y{=}^{\left(d\right)}N\left(0,{3}^{2}+{2}^{2}\right)=N\left(0,13\right)\phantom{\rule{0ex}{0ex}}2X+Y{=}^{\left(d\right)}N\left(0,{4}^{2}+1\right)=N\left(0,5\right)$
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Step 1
We usually say that Ax is a Gaussian vector of length n, that is, for each ${c}_{1},\dots ,{c}_{n}\in \mathbb{R}$, denoting ${Y}_{j}$ the j-th component of Ax, the random variable $\sum _{j=1}^{n}{c}_{j}{Y}_{j}$ has a Gaussian (possibly degenerated) distribution.
Step 2
We usually specify the vector of means of each component and the covariance matrix, whose (i,j) entry is $\mathrm{Cov}\left({Y}_{i},{Y}_{j}\right)$