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If the vector of [[Response variable|response values]] is denoted by <math>\mathbf{y}</math> and the vector of fitted values by <math>\mathbf{\hat{y}}</math>,
:<math>\mathbf{\hat{y}} = \mathbf{P} \mathbf{y}.</math>
As <math>\mathbf{\hat{y}}</math> is usually pronounced "y-hat", the projection matrix <math>\mathbf{P}</math> is also named ''hat matrix'' as it "puts a [[circumflex|hat]] on <math>\mathbf{y}</math>". The formula for the vector of [[errors and residuals in statistics|residual]]s <math>\mathbf{r}</math> can also be expressed compactly using the projection matrix:
:<math>\mathbf{r} = \mathbf{y} - \mathbf{\hat{y}} = \mathbf{y} - \mathbf{P} \mathbf{y} = \left( \mathbf{I} - \mathbf{P} \right) \mathbf{y}.</math>
where <math>\mathbf{I}</math> is the [[identity matrix]]. The matrix <math>\mathbf{M} \equiv \left( \mathbf{I} - \mathbf{P} \right)</math> is sometimes referred to as the '''residual maker matrix'''. Moreover, the element in the ''i''th row and ''j''th column of <math>\mathbf{P}</math> is equal to the [[covariance]] between the ''j''th response value and the ''i''th fitted value, divided by the [[variance]] of the former:
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where <math>\mathbf{\Sigma}</matH> is the [[covariance matrix]] of the error vector (and by extension, the response vector as well). For the case of linear models with [[independent and identically distributed]] errors in which <math>\mathbf{\Sigma} = \sigma^{2} \mathbf{I}</math>, this reduces to:<ref name="Hoaglin1977"/>
:<math>\mathbf{\Sigma}_\mathbf{r} = \left( \mathbf{I} - \mathbf{P} \right) \sigma^{2}</math>.
==Intuition==
[[File:Projection of a vector onto the column space of a matrix.svg|thumb|A matrix, <math>\mathbf{A}</math> has it's column space depicted as the green line. The projection of some vector <math>\mathbf{b}</math> onto the column space of <math>\mathbf{A}</math> is the vector <math>\mathbf{x}</math>]]
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