Normal n-distribution and Variance-covariance matrix

Let x be a standard normal vector of length n.<br.Let $A\in {\mathbb{R}}^{m\times n}$ 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:

$$A=\left[\begin{array}{cc}3& -2\\ 2& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}x{=}^{(d)}N(O,{I}_{2\times 2})=\left[\begin{array}{c}x\\ y\end{array}\right]\phantom{\rule{0ex}{0ex}}Ax=\left[\begin{array}{c}3X-2Y\\ 2X+Y\end{array}\right]$$

$$3X-2Y{=}^{(d)}N(0,{3}^{2}+{2}^{2})=N(0,13)\phantom{\rule{0ex}{0ex}}2X+Y{=}^{(d)}N(0,{4}^{2}+1)=N(0,5)$$

Let x be a standard normal vector of length n.<br.Let $A\in {\mathbb{R}}^{m\times n}$ 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:

$$A=\left[\begin{array}{cc}3& -2\\ 2& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}x{=}^{(d)}N(O,{I}_{2\times 2})=\left[\begin{array}{c}x\\ y\end{array}\right]\phantom{\rule{0ex}{0ex}}Ax=\left[\begin{array}{c}3X-2Y\\ 2X+Y\end{array}\right]$$

$$3X-2Y{=}^{(d)}N(0,{3}^{2}+{2}^{2})=N(0,13)\phantom{\rule{0ex}{0ex}}2X+Y{=}^{(d)}N(0,{4}^{2}+1)=N(0,5)$$