Exercise 5.4.5

Notice that the mean of a linear combination of variables is the linear combination of the means of the variables, i.e. $\overline{\mathit{AX}}=A\overline{X}$.

$Z=E\left[(\mathit{AX}-\overline{\mathit{AX}}){(\mathit{AX}-\overline{\mathit{AX}})}^T\right]=E\left[(\mathit{AX}-A\overline{X}){(\mathit{AX}-A\overline{X})}^T\right]=E\left[A(X-\overline{X}){(X-\overline{X})}^T{A}^T\right]=\mathit{AE}\left[(X-\overline{X}){(X-\overline{X})}^T\right]A^T$

Where in the last step we used the fact that the expectation of linear combination of variables are linear combination of the expectation of variables.