Calculating the variance

Here’s how to prove that $$E\left[\right(y-\mu {)}^{T}A(y-\mu )]=E(y^{T}Ay)-{\mu }^{T}A\mu ,$$

when $A$ is a matrix and $y$ a random vector with mean vector $\mu$.

$$\begin{array}{ccc}E\left[\right(y-\mu {)}^{T}A(y-\mu )]& =& E(y^{T}Ay-{\mu }^{T}Ay-y^{T}A\mu +{\mu }^{T}A\mu )\\ & =& E\left(y^{T}Ay\right)-E\left({\mu }^{T}Ay\right)-E\left(y^{T}A\mu \right)+{\mu }^{T}A\mu \end{array}$$

Here $$E\left({\mu }^{T}Ay\right)={\mu }^{T}AE\left(y\right)={\mu }^{T}A\mu ,$$ by linearity of the expectation operator. Likewise, $$E\left(y^{T}A\mu \right)=E(y{)}^{T}A\mu ={\mu }^{T}A\mu ,$$

and $$E\left(y^{T}Ay\right)-E\left({\mu }^{T}Ay\right)-E\left(y^{T}A\mu \right)+{\mu }^{T}A\mu =E\left(y^{T}Ay\right)-{\mu }^{T}A\mu$$ as claimed.