Sampling from a $2$d normal with a given covariance matrix How would one sample from the $2$-dimensional normal distribution with mean $0$ and covariance matrix $$\begin{bmatrix} a & b\\b & c \end{bmatrix}$$ given the ability to sample from the standard ($1$-dimensional) normal distribution?
This seems like it should be quite simple, but I can't actually find an answer anywhere.

Say you have a random variable $X\sim N(0,E)$ where $E$ is the identity matrix. Let $A$ be a matrix. Then $Y:=AX\sim N(0,AA^T)$. Hence you need to find a matrix $A$ with $AA^T = \left[\matrix{a & b \\ b & c}\right]$. There is no unique solution to this problem. One popular method is the Cholesky decomposition, where you find a triangular matrix $L$ with a given covariance matrix. Another method is to perform a principle axis transform
$$ \left[\matrix{a & b \\ b & c}\right] = U^T\left[\matrix{\lambda_1 & 0 \\ 0 & \lambda_2}\right]U $$
with $UU^T=E$ and then take 
$$ A = U^T\left[\matrix{\sqrt{\lambda_1} & 0 \\ 0 & \sqrt{\lambda_2}}\right]U $$
as a solution. This is the only symmetric positive definite solution then. It is also called the square root of the positive-definite symmetric matrix $AA^T$.
More generally for distribution $X\sim N(\mu,\Sigma)$ it holds $AX+b\sim N(A\mu+b,B\Sigma B^T)$, see Wikipedia.
