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I have to prove that $A^+A$ (where $A^+=\overline{R^t}\left(R\overline{R^t}\right)^{-1}\overline{Q^t}$ is the Moore-Penrose pseudo-inverse of A, with $A=QR$ through QR decomposition) is the orthogonal projection on the columns of A* (A* being the conjugate transpose of A). My problem is: I have no idea where to start. What exactly must I show to prove that it's an orthogonal projection?

Grizzly0111
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  • What are the defining properties of $A^+$? What are the defining properties of an orthogonal projection? In other words, what minimization problem can you relate both of these to? – Ben Grossmann Feb 20 '17 at 16:48

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Since you haven't given us much context here, my best guess for the "right approach" is as follows: using what you know about the pseudoinverse and projections, note that both matrices are such that for a given vector $y$: the output $\hat y = A^+Ay$ (or $\hat y = Q^*(RR^*)^{-1}Q^*y$) is the vector $\hat y = A^*x$ that minimizes $\|A^*x - y\|$.

Ben Grossmann
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