How is the matrix of corrected sum of squares and products defined? In the article Johns, S. "On identifying the population of origin of each observation in a mixture of observations from two normal populations." Technometrics 12.3 (1970): 553-563, the matrix $S$ is defined by the "corrected sum of squares and products of N observations".
Unfortunately I don't understand what is meant by that.
Can someone explain what the exact formula for this matrix is?
 A: This is a somewhat old-fashioned term for the matrices appearing in least squares regression calculations.
The "sum of squares and products" (SSP) matrix can be written $X^\prime X$ where the columns of $X$ are the variables, the rows are the observations, and the entries are the values of those variables for each observation.  The way matrix multiplication works implies that the value in row $i$ and column $j$ of $X^\prime X$ is the vector (dot) product of columns $i$ and $j$ of $X:$ a sum of products.  When $i=j,$ this of course is the sum of squares of column $i.$
The main difference between the SSP matrix and the usual meaning of $X^\prime X$ is that the column of responses, often written as a vector $y,$ is also included in the sum of squares and products matrix.  When there is only one explanatory variable ($X$ is a single column which I will write as "$x$"), this is known as "simple regression." Its SSP matrix often is called $S_{XY}.$   It is a $2\times 2$ matrix,
$$S_{XY} = \pmatrix{\sum_{k} x_k^2 & \sum_{k} x_k y_k \\ \sum_{k} x_k y_k & \sum_{k} y_k^2}.$$
This clearly exhibits the squares and products that are involved.
"Corrected" corresponds to recentering the columns of $X$ before forming its SSP matrix.  The result is a multiple of the usual covariance matrix, $\operatorname{Cov}(X)$ (written $\operatorname{Cov}(x,y)$ in the case of simple regression). The multiple is either $n$ or $n-1,$ depending on what value you divided things by in computing your covariances.
There are many equivalent, well-known formulas for the corrected SSP matrix.  Draper and Smith (at section 1.2) give the following (with the notation updated to reflect a more general symbolism, where $\bar X_i$ is the mean of column $i$ and the matrix has $n$ rows):
$$\begin{aligned}
S_{ij} &= \sum_{k=1}^n (X_{ki} - \bar X_i)(X_{kj} - \bar X_j) \\
&= \sum_{k=1}^n (X_{ki} - \bar X_i)X_{kj} \\
&= \sum_{k=1}^n X_{ki}(X_{kj} - \bar X_j) \\
&= \sum_{k=1}^n X_{ki}X_{kj} - \left(\sum_k X_{ki}\right)\left(\sum_k X_{kj}\right)/n\\
&= \sum_{k=1}^n X_{ki}X_{kj} - n \bar X_i \bar X_j.
\end{aligned}$$
The reason for so many formulas is that until computing devices were commonly available, people had to do these calculations manually.  Some of the formulas are easier to carry out with pencil and paper or handheld calculator, but others are more precise when performed by a digital computer.
Further reading
Draper, Norman Richard, and Harry Smith. 1981. Applied Regression Analysis. Second edition. Wiley Series in Probability and Statistics. New York: Wiley.
https://stats.stackexchange.com/a/108862/919 gives formulas for using the covariance matrix (of all the variables, including $y$) in regression, which is very nearly the same thing as using the corrected SSP matrix.
