I know that the correlation function between random variables $X$ and $Y$ is defined as
$$ \rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E[(X-\mu_X)(Y-\mu_Y)] \over \sigma_X\sigma_Y}. $$
What does happen when $\mathbf{X}$ and $\mathbf{Y}$ are random vectors?
$$ \mathbf{X} = \begin{bmatrix} X_1 \\ X_2 \\ \vdots \\ X_n \end{bmatrix}, \quad \quad \mathbf{Y} = \begin{bmatrix} Y_1 \\ Y_2 \\ \vdots \\ Y_n \end{bmatrix} $$
How is the correlation function defined in this case?
$$ \mathrm{corr}(\mathbf{X},\mathbf{Y}) \, \triangleq \, ? $$