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Problem: Let $\mathcal A$ be an $\mathcal m \times n$ matrix with rank $\mathcal n$ and let $\mathcal A^\dagger$ be its pseudo-inverse. Determine if the operator, $\mathcal AA^\dagger$, is a projection operator. If so what subspace do they project onto?

Correct me where I'm wrong (possibly everywhere): This operator, $\mathcal AA^\dagger$, returns the identity matrix, which should be in the same subspace as $\mathcal A$, right? What would this be?

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    What is your definition of a projection operator? – 2017-02-08
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    @amd I'm not 100% sure. We've only been talking about orthogonal projection operators, does that make sense? – 2017-02-08
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    Most generally, a projection $P$ satisfies $P\circ P=P$. Orthogonal projections have some other properties besides, but that’s the important one to verify. – 2017-02-08
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    @amd Gotcha. So assuming that it returns an identity matrix, then I^2=I, so it is an orthogonal projection? Would this be in R^n as well? – 2017-02-08
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    Be careful. The pseudo-inverse is not In general a right inverse, i.e., $\mathcal A\mathcal A^\dagger\ne I$. This product is, however, the identity on a certain subspace of $\mathbb R^m$ (and that’s a clue to the solution of this problem). – 2017-02-08

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