## 6.5 Useful matrix information in statistics

In statistics, we often work with vectors and matrices. The vector of responses variables is often written as $\textbf{y} = \left[ \begin{array}{c} y_1\\ y_2\\ y_3\\ \vdots\\ y_n \end{array} \right].$ These values of $$y_i$$ are often assumed to have been generated from a process with means $$\mu_i$$, so we can write $\vec{\mu} = \left[ \begin{array}{c} \mu_1\\ \mu_2\\ \mu_3\\ \vdots\\ \mu_n \end{array} \right],$ or $$E[\textbf{y}] = \vec{\mu}$$.

For an appropriately conformable $$m\times n$$ matrix $${C}$$, $$E[C \textbf{y}] = C\vec{\mu}$$. This is the matrix equivalent of $$E[cX] = c E[X] = c\mu$$.

Similarly, $$\text{var}[C\textbf{y}] = C\text{var}[\textbf{y}]C^T$$. This is the matrix equivalent of stating $$\text{var}[c X] = c^2 \text{var}[X]$$.

A further useful thing to note: Consider an unknown $$n\times 1$$ vector $$\textbf{z}$$ and a $$n\times n$$ matrix $$M$$. Then differentiating $$S = \textbf{z}^T M \textbf{z}$$ with respect to $$\textbf{z}$$ gives $\frac{dS}{d\textbf{z}} = 2M\textbf{z}.$ This is the matrix equivalent of differentiating $$y = a x^2$$ and getting $$2ax$$.