$A(A^T A)^{-1} A^T$ is the matrix of the orthogonal projection to im A

$A(A^T A)^{-1} A^T$ is the matrix of the orthogonal projection to im A

Let A in mathbbRntimes m, ker A 0. Show that AAT A-1 AT is the transformation matrix of the orthogonal projection to the image of A. Any advice is greatly appreciated.

Well, its clearly a projection onto the image of A since if yAx is in the image of A then AAT A-1 ATy AAT A-1 ATAx AAT A-1 ATAx Ax y and since clearly it maps anything into the image of A. To show it is orthogonal projection, you need to show it kills anything orthogonal to the image of A. But z is orthogonal to the image of A if zTA0. And in that case also ATz0 and so AAT A-1 ATzAAT A-1 ATz0. The transformation that is the identity on the image of A and kills everything orthogonal to it is the orthogonal projection onto the image of A. So AAT A-1 AT is it. QED

Lets denote M A AT A-1 AT. Then M2 A AT A-1 AT A AT A-1 AT A AT A-1 AT M. And, MT M. These two properties imply that M is the orthogonal projection onto its image. An idempotent operator P is an orthogonal projection iff it is self adjoint imM subset imA should be straightforward to see. Since M is a composition of linear transformations, the last of which is A. To show that imA subset imM, it suffices to show that AT A-1 AT is surjective onto Rm the domain of A. Since AT A-1 is an invertible map, it suffices to show that AT is surjective. In fact, AT is surjective whenever A is injective The transpose of a linear injection is surjective. , so we are done.