Covariance Matrix of Randomly Distributed Vector

Question

If $\mathbf X = (X_1,X_2,...,X_n)^T$ is a randomly distributed vector in $\Bbb R^n$ with mean $μ∈\Bbb R_n$ and covariance matrix $\mathbf C_{XX}$, then let A be an $m×n$ matrix, and define a new random variable $\mathbf Y=A\mathbf X$. Find the covariance matrix $\mathbf C_{YY}$ of $\mathbf Y$ in terms of A and $\mathbf C_{XX}$.

I'm not sure how to go about doing this problem.

Answer 1

Let $\mu=E(X)$. Then $cov(X)=E(XX^T)-\mu\mu^T$ and $cov(AX)=E((AX)(AX)^T)-(A\mu)(A\mu)^T=A(E(XX^T)-\mu\mu^T)A^T=Acov(X)A^T$.