where, e.g., <math>P\{A\} = A \left(A^\top A \right)^{-1} A^\top</math> and <math>M\{A\} = I - P\{A\}</math>.
There are a number of applications of such a partitioning. TheIn the classical application has <math>A</math> is a column of all ones, which allows one to analyze the effects of adding an intercept term to a regression. Another use is in the [[fixed effects model]], where <math>A</math> is a large [[sparse matrix]] of the dummy variables for the fixed effect terms. One can use this partition to compute the hat matrix of <math>X </math> without explicitly forming the matrix <math>X</math>, which might be too large to fit into computer memory.
