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The projection matrix has a number of useful algebraic properties.<ref>{{cite book |last=Gans |first=P. |year=1992 |title=Data Fitting in the Chemical Sciences |url=https://archive.org/details/datafittinginche0000gans |url-access=registration |publisher=Wiley |isbn=0-471-93412-7 }}</ref><ref>{{cite book |last=Draper |first=N. R. |last2=Smith |first2=H. |year=1998 |title=Applied Regression Analysis |publisher=Wiley |isbn=0-471-17082-8 }}</ref> In the language of [[linear algebra]], the projection matrix is the [[orthogonal projection]] onto the [[column space]] of the design matrix <math>\mathbf{X}</math>.<ref name = "Freedman09" /> (Note that <math>\left( \mathbf{X}^\textsf{T} \mathbf{X} \right)^{-1} \mathbf{X}^\textsf{T}</math> is the [[Moore–Penrose pseudoinverse#Full rank|pseudoinverse of X]].) Some facts of the projection matrix in this setting are summarized as follows:<ref name = "Freedman09" />
* <math>\mathbf{u} = (\mathbf{I} - \mathbf{P})\mathbf{y},</math> and <math>\mathbf{u} = \mathbf{y} - \mathbf{P} \mathbf{y} \perp \mathbf{X}.</math>
* <math>\mathbf{P}</math> is symmetric, and so is <math>\mathbf{M} \equiv
* <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=[https://archive.org/details/advancedeconomet00amem/page/460 460]–461 |url=https://archive.org/details/advancedeconomet00amem |url-access=registration }}</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>.
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