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I am trying to solve this old exam question: Show that $P = A(A^TA)^{-1}A^T$ is a projection matrix if $A = \begin{bmatrix}1 \\ m\end{bmatrix}$. I don't understand what I'm doing wrong here: $P = A(A^{-1}{A^T}^{-1}) A^T \implies P = (AA^{-1})({A^T}^{-1} A^T) \implies P = I$

However, if I actually multiply it out, I do get the projection matrix. Obviously $I$ is not a projection matrix, but I'm not sure what I'm doing wrong to get the identity though.

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    hesso$n$: A projection matrix is invertible if and only if it is the identity matrix. What is the definition of projection matrix?2012-12-20

2 Answers 2

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Billy has pointed out a problem with your approach, but here is a suggestion for one way to see why $P$ is a projection. This does not depend on the particular form of $A$, as long as $(A^\text{T}A)^{-1}$ exists (which it does in your case). Then you have

$ \left[A(A^\text{T}A)^{-1}A^\text{T}\right]\left[A(A^\text{T}A)^{-1}A^\text{T}\right]= A\left[(A^\text{T}A)^{-1}(A^\text{T}A)\right](A^\text{T}A)^{-1}A^\text{T}.$

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$A = \begin{bmatrix}1 \\ m\end{bmatrix} $ $\Rightarrow $ $A(A^TA)^{-1}A^T$ = $ \frac{1}{m^2+1}\begin{bmatrix}1 & m \\ m & m^{2}\end{bmatrix} \Rightarrow $ $ (A(A^TA)^{-1}A^T)^{2} = $ $ (\frac{1}{m^2+1})^{2}\begin{bmatrix}1 & m \\ m & m^{2}\end{bmatrix}\begin{bmatrix}1 & m \\ m & m^{2}\end{bmatrix} = $ $ (\frac{1}{m^2+1})^{2}\begin{bmatrix}1 + m^{2}& m + m^3 \\ m + m^3 & m^{2} + m^{4}\end{bmatrix} = $ $ (\frac{1}{m^2+1})\begin{bmatrix}1& m \\ m & m^2\end{bmatrix} = (A(A^TA)^{-1}A^T) $ Therefore $P$ is a projection matrix. Since $P = (A(A^TA)^{-1}A^T) = (A(A^TA)^{-1}A^T)^{2} = P^{2} $

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    $I$ is a projection matrix because $I^{2}$ = $I$2012-12-20