Finding covariance from a transformed random variable given its covariance matrix

Question

Let the bivariate random variable $A=(A_1,A_2)^T$ have a Gaussian distribution on $\mathbb{R}^2$ with zero mean and covariance matrix be given by

$$\begin{pmatrix} 1 & -0.4\\-0.4 & 1\end{pmatrix}$$.

Let $B$ = $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $C$= $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$. Define $X=B^TA,Y=C^TA$. How do I find the covariance of X and Y?

I know that $cov(X,Y) = E(XY)-E(X)E(Y)$. I don't quite understand how to read a covariance matrix.

Answer 1

You have $B^\top A = A_1 + 2A_2$ and $C^\top A = 2A_1 + A_2$. Then,

\begin{eqnarray*} Cov\left(X,Y\right) &=& Cov\left(A_1+2A_2,2A_1+A_2\right)\\ &=& 2Cov\left(A_1,A_1\right) + Cov\left(A_1,A_2\right) + 4Cov\left(A_1,A_2\right)+2Cov\left(A_2,A_2\right). \end{eqnarray*}

Answer 2

You can use matrix properties:

$$\begin{split}\text{Cov}(B^TA, C^TA)&=B^T\text{Cov}(A, A)C\\ &=B^T\text{Var}(A)C\end{split}$$

It is 1-dimensional (a scalar).