Sure, yes for such a simple example, where the matrix is proportional to a projector.

But it is not obvious what would such a generalized inverse be for more elaborated examples like this singular quadratic matrix , or a non-quadratic matrix . The Moore-Penrose inverse is one specific prescription to get the unique “inverse” as the solution of the defining conditions 1-4. For some simple classes of matrices one can get some simplified expressions for their MP inverses (as displayed in the main body of this posting), but for the generic cases one has to do homework and solve conditions 1-4 afresh.

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