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Math Help - Product of 2 projection matrices

  1. #1
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    Product of 2 projection matrices

    So, I'm working on a proof where if I had product of projection matrices Pi and Pj equal zero, the proof works. The problem statement is kind of long, so I won't write it out..

    My question is "does a product of two projection matrices ever equal zero??" If it does, in what cases? TIA!
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  2. #2
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    Quote Originally Posted by borracho View Post
    So, I'm working on a proof where if I had product of projection matrices Pi and Pj equal zero, the proof works. The problem statement is kind of long, so I won't write it out..

    My question is "does a product of two projection matrices ever equal zero??" If it does, in what cases? TIA!
    Well, there's a lot you can say. The first things which pop into my mind are as follows: Suppose for two projections P,Q on \mathsf{V} we have PQ=QP=0. Then (P+Q)^2=P^2+Q^2+PQ+QP=P+Q. So P+Q is also a projection. Furthermore, \mathsf{R}(Q)\subset\mathsf{N}(P) and \mathsf{R}(P)\subset\mathsf{N}(Q), such that \text{rank }P+\text{rank }Q\leq \dim\mathsf{V} and \dim\mathsf{V}\leq\text{nullity }P+\text{nullity }Q.
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  3. #3
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    If P projects vectors onto subspace U, Q projects vectors onto subspace V and U and V are orthogonal subspaces, then PQ= QP= 0.
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