Orthogonal Projections: Same Thing or Not?

[Isaac0427]

TL;DR Summary

Why is the orthogonal projection formula written as $P_A=A(A^TA)^{-1}A^T$ as opposed to $P_A=(A^T)^{-1}A^T$?

Aren't they the same thing? If so, why would textbooks write the former? Ex: https://textbooks.math.gatech.edu/ila/projections.html or http://www.math.lsa.umich.edu/~speyer/417/OrthoProj.pdf or https://en.wikipedia.org/wiki/Projection_(linear_algebra)#Orthogonal_projections

Thank you!

 

[fresh_42]

$\left(A^\tau\right)^{-1}\left(A\right)^\tau=1$ which doesn't mean a lot. $A\left(A^\tau A\right)^{-1}A^\tau=1$ which is also meaningless. A projection is generally a transformation for which $A^2=A$ holds.

 

[Infrared]

You can't simplify $(A^TA)^{-1}=A^{-1}(A^T)^{-1}$ since $A$ is not a square matrix.

@fresh_42 I think $A$ is a matrix whose columns form a basis for the subspace of the projection, not the projection matrix itself.

 

[Isaac0427]

You can't simplify (ATA)−1=A−1(AT)−1 since A is not a square matrix.

Ah, that's what I was missing. Ok, thank you.

 

[Isaac0427]

@fresh_42 I think A is a matrix whose columns form a basis for the subspace of the projection, not the projection matrix itself.

And yes. I see though that it would all be meaningless if they were invertible matrices, as then the span of A would be all of $\mathbb{R}^n$, and a vector in $\mathbb{R}^n$'s projection onto $\mathbb{R}^n$ is just itself.