# What is the variance of Y = AX where A is a matrix?

Here $$X,Y$$ are vector of length $$n$$ and $$A$$ is a $$n\times n$$ matrix. Suppose the covariance matrix $$D(X)$$ is known?

This (linear transform) is typically listed as a property of covariance, but easy to show as well: \begin{align}\operatorname{cov}(AX)&=\mathbb E[AXX^TA^T]-\mathbb E[AX]\mathbb E[X^TA^T]\\&=A\mathbb E[XX^T]A^T-A\mathbb E[X]\mathbb E[X^T]A^T\\&=A(\mathbb E[XX^T]-\mathbb E[X]\mathbb E[X^T])A^T\\&=ADA^T\end{align}