Block matrix

In general we can partition the $k$ characteristics contained in a $p\times 1$ random vector $X$ into several groups of arbitrary sizes.

$$\begin{array}{rl}X& =\left[\begin{array}{c}{X}_1\\ X_2\\ X_3\\ ⋮\\ X_n\end{array}\right]\end{array}$$

Assume that $X_1\dots X_q$ belong to personal traits and $X_{p+1}\dots X_n$ belong to physical traits. Then we can denote $X$ as:

$$X={\left[\begin{array}{c}{X}_{(1),q\times 1}\\ X_{(2),n-q\times 1}\end{array}\right]}_{n\times 1}$$

Then the expectation is:

$$\mu =E[X]=\left[\begin{array}{c}{\mu }_1\\ ⋮\\ {\mu }_q\\ {\mu }_{q+1}\\ ⋮\\ {\mu }_n\end{array}\right]=\left[\begin{array}{c}{\mu }_{(1)}\\ {\mu }_{(2)}\end{array}\right]$$

and the $\Sigma$ is

$${\Sigma }_{n\times n}=E[(X-\mu {)}_{n\times 1}(X-\mu {)}_{1\times n}^T]$$ $$\Sigma ={\left[\begin{array}{cc}{\Sigma }_{(11)}^{q\times q}& {\Sigma }_{(12)}^{q\times p-q}\\ {\Sigma }_{(21)}^{p-q\times q}& {\Sigma }_{(22)}^{p-q\times p-q}\end{array}\right]}_{n\times n}$$