The Exercise

Two systems have finite-dimensional state spaces $V$ and $W$, and the composite system has the state space $V \otimes W$. Let $P$ be a projector on $V$ and $Q$ a projector on $W$. The exercise asks to show that $P \otimes Q$, which acts as $P$ on the first factor and as $Q$ on the second, is a projector on $V \otimes W$.

Projectors and the Kronecker Product

A projector $A$ is a Hermitian square matrix that equals its own square: $$A^2 = A, \qquad A^\dagger = A,$$(1) where $A^\dagger$ is the conjugate transpose of $A$.

The Kronecker product $A \otimes B$ of a square matrix $A$, with rows and columns labelled by a finite set $I$, and a square matrix $B$, labelled by a finite set $J$, is the square matrix labelled by the pairs in $I \times J$ with entries $$(A \otimes B)_{(i,k),(j,l)} = A_{ij}\, B_{kl}.$$(2)

The Claim

With $A$ and $B$ the matrices of $P$ and $Q$ in bases of $V$ and $W$, the Kronecker product is the matrix of $P \otimes Q$, and this is the exercise.

What Is Checked

The claim is stated over square matrices with entries in any commutative semiring with a conjugation, a set with addition and multiplication obeying the usual rules but not generally subtraction, and $A^\dagger$ in (1) is the transpose of $A$ with each entry sent through that conjugation; complex matrices under complex conjugation are one case. Over all of these, the library proves that the Kronecker product (2) takes a pair of projectors to a projector.^✦