I am reading this book and came across a transformation that I did not understand:
$\frac{1}{N} \sum_{i=1}^{N} (\mathbf{x}_{i} - \mathbf{\bar{x}}) (\mathbf{x}_{i} - \mathbf{\bar{x}})^{T} = \frac{1}{N} (\sum_{i=1}^{N} \mathbf{x}_{i}\mathbf{x}_{i}^{T})-\mathbf{\bar{x}}\mathbf{\bar{x}}^{T}$
where $\mathbf{x}_{i}$ and $\mathbf{\bar{x}}$ are both vectors.
When I expand the brackets I get something like this:
$\frac{1}{N} \sum_{i=1}^{N} (\mathbf{x}_{i} - \mathbf{\bar{x}}) (\mathbf{x}_{i} - \mathbf{\bar{x}})^{T} = \frac{1}{N} \sum_{i=1}^{N} \mathbf{x}_{i}\mathbf{x}_{i}^{T}-\mathbf{x}_{i}\mathbf{\bar{x}}^{T}-\mathbf{\bar{x}}\mathbf{x}_{i}^{T}+\mathbf{\bar{x}}\mathbf{\bar{x}}^{T}$
Can someone explain to me what I am doing wrong?
You are not doing anything wrong \begin{eqnarray*} \frac{1}{N} \sum_{i=1}^{N} (\mathbf{x}_{i} - \mathbf{\bar{x}}) (\mathbf{x}_{i} - \mathbf{\bar{x}})^{T} = \frac{1}{N} \sum_{i=1}^{N} (\mathbf{x}_{i}\mathbf{x}_{i}^{T}-\mathbf{x}_{i}\mathbf{\bar{x}}^{T}-\mathbf{\bar{x}}\mathbf{x}_{i}^{T}+\mathbf{\bar{x}}\mathbf{\bar{x}}^{T}). \end{eqnarray*} Now use $ \mathbf{\bar{x}}= \frac{1}{N} \sum_{i=1}^{N} \mathbf{x}_{i} $ \begin{eqnarray*} \frac{1}{N} \sum_{i=1}^{N} (\mathbf{x}_{i} - \mathbf{\bar{x}}) (\mathbf{x}_{i} - \mathbf{\bar{x}})^{T} = \frac{1}{N} \sum_{i=1}^{N} (\mathbf{x}_{i}\mathbf{x}_{i}^{T})-\mathbf{\bar{x}}\mathbf{\bar{x}}^{T}-\mathbf{\bar{x}}\mathbf{\bar{x}}^{T}+\mathbf{\bar{x}}\mathbf{\bar{x}}^{T} \end{eqnarray*} so the terms cancel out and you will obtain the desired result.