Suppose I have the matrix given by:
$$X = \begin{bmatrix}1 & 0 & 0\\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$$
This matrix actually represents whether a user interacted with a game on day 1, 2 or 3. The rows are users and days are columns.
Next, we subtract the column means from this matrix:
$$\hat{X} = X - colMean(X) = \begin{bmatrix} 0 & -0.66 & -0.33 \\ 0 & 0.33 & -0.33 \\ 0 & 0.33 & 0.66 \end{bmatrix}$$
My questions are:
Is there any significance to the product:
$$\hat{X}^TX?$$
The $colMean(X)$ represents the retention rate of the users who interact with my game and the curve produced follows a power law. $X$ also has some covariance matrix, $\Sigma$. Suppose, I gather more information about users and their interaction with my game on the first three days. I would like to use $\mu = colMean(X)$ and the covariance, $\Sigma$, as a prior and then update those parameters with new observations.
How would I go about doing this?
Any comments or references would be greatly appreciated. I just need an idea of where to start. Thank you in advance.