# Expectation of a constant matrix multiplied with a random vector and its transpose?

It is given that a random vector $$\mathbf{y} \sim N(\mathbf{0}, \boldsymbol{\Theta})$$, and $$\mathbf{A}$$ is a constant matrix. Assuming that the dimensions are compatible, what would $$E(\mathbf{A} \mathbf{y} \mathbf{y}^T)$$ equal?

I understand that $$E(\mathbf{A} \mathbf{y}) = \mathbf{A} E(\mathbf{y})$$ and that $$E(\mathbf{y} \mathbf{y}^T)$$ is simply the covariance-variance matrix of the random vector. But I am unable to evaluate the above expression. Any help would be much appreciated; thanks for reading!

• I understand that $E(\mathbf{A} \mathbf{y}) = \mathbf{A} E(\mathbf{y}) \mathbf{A}^T \quad\quad$No, : $Var(Ay) = A Var(y) A^T$. Your answer below is correct. Jul 3 at 13:57
• You're right, thanks. Jul 3 at 15:10

Turns out that the expression evaluates to $$\mathbf{A} E(\mathbf{y} \mathbf{y}^T),$$ which is just $$\mathbf{A} \mathbf{\Theta}$$.