Pearson correlation coefficient

$$P={\left[\begin{array}{cccc}1& {\rho }_{12}& \dots & {\rho }_{1p}\\ {\rho }_{21}& 1& \dots & {\rho }_{2p}\\ ⋮& ⋮& \ddots & {\rho }_{3p}\\ {\rho }_{p1}& \dots & \dots & 1\end{array}\right]}_{p\times p}$$

where,

$${\rho }_{ij}=\frac{Cov(x_i,x_j)}{\sqrt{V(x_i)}\sqrt{V(x_j)}}=\frac{{\sigma }_{12}}{\sqrt{{\sigma }_{11}}\sqrt{{\sigma }_{22}}}$$

observe that

$${\rho }_{ii}=\frac{{\sigma }_{11}}{\sqrt{{\sigma }_{11}^2}}=1$$

R is symmetric matrix.

$R$ measures Linear Association. If $r_{ij}=0$, it means that there is no linear association but there might be any non-linear effect.

R doesn't have any units of measurement and $r_{ij}\in [-1,1]$. And it is invariant under linear transformation.

$$\begin{array}{rl}{y}_{ij}& =a{x}_{ij}+b\\ ⟹cor(y_i,y_j)& =cor(x_i,x_j)\end{array}$$