# Variance of sum of linear combination

I want to calculate the variance of a sum of linear combinations, so $$\operatorname{Var}\left(w'R_1 + w'R_2\right)$$ where $w$ is a $N\times 1$ vector and both $R_1$ and $R_2$ are $N\times 1$ vectors as well. I have a cross autocorrelation matrix $Q$ which is a $N\times N$ matrix. Now the covariance matrix of $R_i$ is denoted by $\Sigma_i$. Uhm I guess $\operatorname{Var}(w'R_i)=w'\Sigma_iw$.

Here is the thing $R_1$ and $R_2$ are correlated, hence the $Q$ matrix. How to calculate the above mentioned variance component?

• As far as I'm aware there's no such thing as a cross autocorrelation matrix. Do you mean the cross-correlation matrix between $R_1$ and $R_2$? – joriki Jun 30 '16 at 16:35
• Well with a cross autocorrelation matrix I mean something like in the pdf here, please see slide 30 lipas.uwasa.fi/~sjp/Teaching/Afts/Lectures/etsc22.pdf – Eren Jun 30 '16 at 16:36
• So $Q_{ij}$ gives the correlation between $R_{1i}$ and $R_{2j}$ is that how we interpret it? – user237392 Jun 30 '16 at 18:58
• Yes that is how you interpret it, but how is it a covariance quantity? I mean the $Q$ matrix is standardized by its variance to become an autocorrelation matrix and thus it is a correlation quantity. So $i$-th element of $R_1$ is correlated with the $j$-th element of $R_2$ by the term $Q_{ij}$. – Eren Jun 30 '16 at 19:43