Consider this 4x3 matrix.

Now consider the matrix product, X'X.

The result (product) is a square matrix, whose elements are,

The diagonal values are sums of squares and the off-diagonal values are sums of cross products. The matrix is an SSCP (Sums of Squares and Cross Products) matrix. Thus

Anytime you see the matrix notation X'X or D'D or Z'Z, the resulting product will be a SSCP matrix.

Sometimes, you will see this notation, 1/(n-1)D'D or 1/(n-1)Z'Z, this is just an SSCP that has been adjusted by having every element divided by the constant n-1. This is sometimes called an average SSCP matrix.

The matrix 1/(n-1)D'D is also known as the variance-covariance matrix.

The matrix 1/(n-1)Z'Z is also known as the correlation matrix.

Examples in Stata and Mata.

Stata

mat X = (4,2,2 \ 4,6,8 \ -2,2,4)

mat list X

X[3,3]
    c1  c2  c3
r1   4   2   2
r2   4   6   8
r3  -2   2   4

mat sscp = X'*X

mat list sscp

symmetric sscp[3,3]
    c1  c2  c3
c1  36
c2  28  44
c3  32  60  84

: X = (4,2,2 \ 4,6,8 \ -2,2,4)

: X
        1    2    3
    +----------------+
  1 |   4    2    2  |
  2 |   4    6    8  |
  3 |  -2    2    4  |
    +----------------+

: sscp = X'*X

: sscp

[symmetric]
        1    2    3
    +----------------+
  1 |  36            |
  2 |  28   44       |
  3 |  32   60   84  |
    +----------------+

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