# Reading and interpreting the scatter matrix [duplicate]

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?
• (1) That is correct. (2) See. Scatter is covariance without the denominator. 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?
– user35349
Jan 28 '14 at 11:47
• Yeah, 2 names. The scatter, = The (summed) cross-product of centered variables. Jan 28 '14 at 11:59
• 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.
– whuber
Jan 28 '14 at 13:56