# Hat Matrix Projection

**1A square matrix A is a projection if it is idempotent 2A projection A is orthogonal if it is also symmetric.**

**Hat matrix projection**.
The hat matrix H is defined in terms of the data matrix X.
P P.
How can we prove that from first principles ie.

It is a bit more convoluted to prove that any idempotent matrix is the projection matrix for some subspace but thats also true. San 325 655-2060 325 Angelo Motor Co 450-1052. SAN ANGELO MOTOR CO.

Therefore we have P2 H C P. First and most simply the projection matrix is square. These two conditions can be re-stated as follows.

I understand that the trace of the projection matrix also known as the hat matrix XInv XXX in linear regression is equal to the rank of X. As before we do not have a solution to so instead we look to solve. The residual vector is given by e In Hy with the variance-covariance matrix V In Hσ2 where In is the identity matrix of order n.

Projection matrices are also symmetric ie. The hat matrix H is the projection matrix that expresses the values of the observations in the independent variable y in terms of the linear combinations of the column vectors of the model matrix X which contains the observations for each of the multiple variables you are regressing on. In fact you can show that since P is symmetric and idempotent it satisfies 0.

21 Residuals The vector of residuals e is just e y x b 42 Using the hat matrix e y Hy I H. Call this matrix the hat matrix because it puts the hat on. H X XTX 1XT and determines the fitted or predicted values since.