Need help with proving that $E[x_n, x_m] = \mu\mu^T +I_{nm}Σ$

Question

Need to use the following properties:

E(x) = ${\mu}$

E(${x}{x} ^ {T})=μ {μ } ^ {T} + Σ$

To prove that :

$E[x_n, x_m] = \mu\mu^T +I_{nm}Σ$

$x_n,x_m$ - are vectors

${\mu}$ - mean

Σ - covariance matrix

$I_{nm}$ is identity matrix with vectors $x_n,x_m$ as column vectors

Not quite sure where to start here.

Answer 1

$E(xx^\top) = E([x-\mu + \mu][x^\top - \mu^\top + \mu^\top])$. Now multiply out everything that is inside the expectation, treating $x-\mu$ and $x^\top - \mu^\top$ as individual terms. Use linearity of the expectation and the definition of the covariance matrix.