Thread: Product of 2 projection matrices

1. 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!

2. Originally Posted by borracho
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 $\dim\mathsf{V}\leq\text{nullity }P+\text{nullity }Q$.

3. If P projects vectors onto subspace U, Q projects vectors onto subspace V and U and V are orthogonal subspaces, then PQ= QP= 0.