As a part of conducting a likelihood ratio test we've defined $W$. $X$ is a $N_p$-distributed r.v. and $\bar{X}$ is vector of the estimated means.
$$W = \sum_{i = 1}^{n} (X_i - \bar{X})(X_i - \bar{X})^T $$
And I have to proof that the trace of $W$ is:
$$\text{tr}(W) = \sum_{i = 1}^{n} X_i^T X_i + n\text{tr} \bar{X}\bar{X}^T$$
I've tried the following:
$$\text{tr}(W) = \text{tr} \left(\sum_{i = 1}^{n} (X_i - \bar{X}) (X_i - \bar{X})^T \right)$$ $$\text{tr}(W) = \sum_{i = 1}^{n} \text{tr} ((X_i - \bar{X})^T (X_i - \bar{X}))$$ $$\text{tr}(W) = \sum_{i = 1}^{n} \text{tr}( (X_i^T - \bar{X}^T) (X_i - \bar{X}))$$ $$\text{tr}(W) = \sum_{i = 1}^{n} \text{tr}(X_i^TX_i -X_i^T\bar{X} - \bar{X}^TX_i + \bar{X}^T\bar{X} )$$ $$\text{tr}(W) = \sum_{i = 1}^{n} \text{tr}( X_i^TX_i -X_i^T\bar{X} - \bar{X}^TX_i + \bar{X}^T\bar{X} )$$ $$\text{tr}(W) = \sum_{i = 1}^{n} (X_i^TX_i - X_i^T\bar{X} - \bar{X}^TX_i) + n \text{tr} \bar{X}^T\bar{X} $$
But I can't figure out how to get the $X_i^T\bar{X} - \bar{X}^TX_i$ to "disappear". Any help is appreciated!