Exercise 2.32

Suppose \(P_{V'}\) is a projector in vector space \(V\) and \(P_{W'}\) is a projector in vector space \(W\). The tensor product of them is given by \(P_{V'} \otimes P_{W'} \). To prove \(P_{V'} \otimes P_{W'} \) is projector, we only need to prove \(P_{V'} \otimes P_{W'} \) is Hermitian and \((P_{V'} \otimes P_{W'} )^2 = P_{V'} \otimes P_{W'} \).

From Exercise 2.30 we know that since projector \(P_{V'}\) and \(P_{W'}\) are Hermitian, so \(P_{V'} \otimes P_{W'}\) is also a Hermitian operator. Meanwhile, since we have

(1)\[ (P_{V'} \otimes P_{W'})^2 = (P_{V'} \otimes P_{W'})(P_{V'} \otimes P_{W'})^2 = P_{V'}P_{V'} \otimes P_{W'}P_{W'} \]

and for projector \(P_{V'}\) and \(P_{W'}\) we have \(P_{V'}^2 = P_{V'}\) and \(P_{W'}^2 = P_{W'}\), then we could have \((P_{V'} \otimes P_{W'})^2 = P_{V'} \otimes P_{W'}\). Therefore, we can conclude from the above statement that, the tensor product of two projectors is a projector.