Projection matrix: Difference between revisions

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== Blockwise formula ==

Suppose the design matrix <math>X</math> can be decomposed by columns as <math>X= [A, ~~~B]</math>.

Define the hat or projection operator as <math>P\{X\} = X \left(X^\top X \right)^{-1} X^\top</math>. Similarly, define the residual operator as <math>M\{X\} = I - P\{X\}</math>.

Then the projection matrix can be decomposed as follows:<ref>{{cite book|last1=Rao|first1=C. Radhakrishna|last2=Toutenburg|first2=Helge|author3=Shalabh|first4=Christian|last4=Heumann|title=Linear Models and Generalizations|year=2008|publisher=Springer|___location=Berlin|isbn=978-3-540-74226-5|pages=323|edition=3rd}}</ref>