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Thread: 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 $\displaystyle P,Q$ on $\displaystyle \mathsf{V}$ we have $\displaystyle PQ=QP=0$. Then $\displaystyle (P+Q)^2=P^2+Q^2+PQ+QP=P+Q$. So $\displaystyle P+Q$ is also a projection. Furthermore, $\displaystyle \mathsf{R}(Q)\subset\mathsf{N}(P)$ and $\displaystyle \mathsf{R}(P)\subset\mathsf{N}(Q)$, such that $\displaystyle \text{rank }P+\text{rank }Q\leq \dim\mathsf{V}$ and $\displaystyle \dim\mathsf{V}\leq\text{nullity }P+\text{nullity }Q$.
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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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