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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?
ttnphns
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  • (1) That is correct. (2) [See](http://stats.stackexchange.com/a/22520/3277). Scatter is covariance without the denominator. – ttnphns Jan 28 '14 at 11:20
  • So $s_{jk}$ is the covariance between variable $j$ and $k$ multiplied by (n-1)? What would that mean, @ttnphns? Has that a name? –  Jan 28 '14 at 11:47
  • Yeah, 2 names. The scatter, = The (summed) cross-product of centered variables. – ttnphns Jan 28 '14 at 11:59
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    A nearly identical question concerning $S/n$ (or $S/(n-1)$ as some prefer), which is the *covariance matrix*, is addressed at http://stats.stackexchange.com/questions/18058/how-would-you-explain-covariance-to-someone-who-understands-only-the-mean/18200#18200. If that doesn't suit you, then please search our site for more information on [covariance matrices](http://stats.stackexchange.com/search?q=covariance+matrix). For direct references to scatter matrices, search on [that term](http://stats.stackexchange.com/search?q=%22scatter+matrix%22). – whuber Jan 28 '14 at 13:56

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