Projection matrix: Difference between revisions

Revision as of 02:39, 1 August 2023

Revision as of 02:39, 1 August 2023

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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:= \mathbf{I} - \mathbf{P}</math> is sometimes referred to as the '''residual maker matrix''' or the '''annihilator matrix'''.

The [[covariance matrix]] of the residuals <math>\mathbf{r}</math>, by [[error propagation]], equals