The scatter matrix is defined as

$$S = \sum_{j=1}^n (\mathbf{x}_j-\overline{\mathbf{x}})(\mathbf{x}_j-\overline{\mathbf{x}})^T$$

The trace (sum of the diagonal elements) of this matrix is equivalent to the overall sum of squares.

  1. Each element $s_{jj}$ of the diagonal is equivalent to the sum of squares (of deviations) of the $j$-th variable, is that correct?
  2. How about the other elements $s_{jk}$ where $j \ne k$ ? What do they represent and thus mean?
  • $\begingroup$ (1) That is correct. (2) See. Scatter is covariance without the denominator. $\endgroup$
    – ttnphns
    Jan 28, 2014 at 11:20
  • $\begingroup$ So $s_{jk}$ is the covariance between variable $j$ and $k$ multiplied by (n-1)? What would that mean, @ttnphns? Has that a name? $\endgroup$
    – user35349
    Jan 28, 2014 at 11:47
  • $\begingroup$ Yeah, 2 names. The scatter, = The (summed) cross-product of centered variables. $\endgroup$
    – ttnphns
    Jan 28, 2014 at 11:59
  • 1
    $\begingroup$ A nearly identical question concerning $S/n$ (or $S/(n-1)$ as some prefer), which is the covariance matrix, is addressed at stats.stackexchange.com/questions/18058/…. If that doesn't suit you, then please search our site for more information on covariance matrices. For direct references to scatter matrices, search on that term. $\endgroup$
    – whuber
    Jan 28, 2014 at 13:56