Guys I'm stuck on a question in my textbook. Let me first tell you about the question before I explain what I don't understand.
Let $x:\Omega \to \mathbb{R}^n$, $x\in G(0,Q_x)$, $w:\Omega\to\mathbb{R}^k$,$w\in G(0,Q_w)$, $Q_w = Q^T_w > 0$, $C\in\mathbb{R}^{k \times n}$ (where G means Gaussian random variable with mean and variance as specified). Assume that $x,w$ are independent random variables. Define $y:\Omega\to\mathbb{R}^k$ by
$y = Cx+w$.
Determine an expression for $E\left[x|\mathcal{F}^y\right]$, and for $E\left[e^{iu^T x}|\mathcal{F^y}\right]$.
By two different prepositions from my textbook I found that:
$y\in G\left(0,CQ_x C^T+Q_w\right)$. So let's call $Q_y = CQ_x C^T+Q_w$. Now I don't understand how to calculate the cross covariance matrix $Q_{xy}$. Because once I find $Q_{xy}$ then I know that:
$E\left[x|\mathcal{F^y}\right] = m_x + Q_{xy}Q_y^{-1} \left( y-m_y \right) = Q_{xy}Q_y^{-1}y = Q_{xy}Q_y^{-1}\left(Cx+w\right)$
and
$E\left[e^{iu^T x}|\mathcal{F^y}\right] = e^{iu^T E\left[x|\mathcal{F^y}\right] - \frac{1}{2} u^T \tilde{Q}u}$, where $\tilde{Q} = Q_x-Q_{xy}Q_y^{-1}Q_{xy}^T$
(given by $E\left[\left(x-E\left[x|\mathcal{F^y}\right]\right)\left(x-E\left[x|\mathcal{F^y}\right]\right)^T\right] = \tilde{Q}$)
Note that since all involved Gaussians have "unconditional" mean zero, the covariance matrix of $x$ and $y$ is given by $$\mathbf{E}[yx']=C \mathbf{E}[xx']+ \mathbf{E}[wx'] = C \mathbf{E}[xx']=CQ_{xx}$$ where I used that $x$ and $w$ are independent. The apastrophe ' denotes the transpose operator.