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Orthogonal Projection operators

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Let P1 be the orthogonal projector onto the subspace $1, P2 the orthogonal projector onto the subspace$2. Show that, for the product P1P2 to be an orthogonal projector as well, it is necessary and sufficient that P1 and P2 commute. In this case, what is the subspace onto which P1P2 projects?

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Solution Summary

The solution demonstrates how to show that if two projectors commute, their product is also an orthogonal projector.