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_{v}(x) = A(A

^{T}A)

^{-1}A

^{T}x

I'm puzzled why this equation doesn't reduce to Proj

_{v}(x) = IIx

since (A

^{T}A)

^{-1}= A

^{-1}(A

^{T})

^{-1}so that should mean that A(A

^{T}A)

^{-1}A

^{T}= AA

^{-1}(A

^{T})

^{-1}A

^{T}= II

What is wrong with my reasoning?

Thanks.