Projection matrix: Difference between revisions

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* <math>\mathbf{P}</math> is idempotent: <math>\mathbf{P}^2 = \mathbf{P}</math>, and so is <math>\mathbf{M}</math>.

* If <math>\mathbf{X}</math> is an {{nowrap|''n'' × ''r''}} matrix with <math>\operatorname{rank}(\mathbf{X})=r</math>, then <math>\operatorname{rank}(\mathbf{P}) = r</math>

* The [[eigenvalue]]s of <math>\mathbf{P}</math> consist of ''r'' ones and {{nowrap|''n'' − ''r''}} zeros, while the eigenvalues of <math>\mathbf{M}</math> consist of {{nowrap|''n'' − ''r''}} ones and ''r'' zeros.<ref>{{cite book |first=Takeshi |last=Amemiya |title=Advanced Econometrics |___location=Cambridge |publisher=Harvard University Press |year=1985 |isbn=0-674-00560-0 |pages=460–461[https://archive.org/details/advancedeconomet00amem/page/460 460]–461 |url=https://booksarchive.google.comorg/books?id=0bzGQE14CwEC&pgdetails/advancedeconomet00amem |url-access=PA460registration }}</ref>

* <math>\mathbf{X}</math> is invariant under <math>\mathbf{P}</math> : <math>\mathbf{P X} = \mathbf{X},</math> hence <math>\left( \mathbf{I} - \mathbf{P} \right) \mathbf{X} = \mathbf{0}</math>.

* <math>\left( \mathbf{I} - \mathbf{P} \right) \mathbf{P} = \mathbf{P} \left( \mathbf{I} - \mathbf{P} \right) = \mathbf{0}.</math>