Understanding the Limitations of the Projection onto a Subspace Equation

[fredsmithsfc]

The Proj_v(x) = A(A^TA)^-1A^Tx

I'm puzzled why this equation doesn't reduce to Proj_v(x) = IIx

since (A^TA)^-1 = A^-1(A^T)^-1 so that should mean that A(A^TA)^-1A^T = AA^-1(A^T)^-1A^T = II

What is wrong with my reasoning?

Thanks.

 

[Fredrik]

It doesn't look wrong to me. Where did you get the idea that the first expression is a projection operator?

 

[monea83]

The problem is that A will be rectangular (non-square) if you are projecting onto a subspace, and thus its inverse does not exist (e.g. A is a column vector for projection onto a line).

 

[fredsmithsfc]

It doesn't look wrong to me. Where did you get the idea that the first expression is a projection operator?

Hi Fredrik,

I first saw it in the Khan Academy Linear Algebra video: "Lin Alg: A Projection onto a Subspace is a Linear Transformation" which is at this link: http://www.khanacademy.org/video/lin-alg--a-projection-onto-a-subspace-is-a-linear-transforma?playlist=Linear%20Algebra

But I also found it at Wikipedia here: http://en.wikipedia.org/wiki/Projection_(linear_algebra)

and in the book "Matrix Analysis and Applied Linear Algebra" by Carl Meyer on page 430

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[fredsmithsfc]

The problem is that A will be rectangular (non-square) if you are projecting onto a subspace, and thus its inverse does not exist (e.g. A is a column vector for projection onto a line).

That makes sense. Thanks.