Is $X\otimes X = (X\otimes I)(I\otimes X)$?

Question

Does $X\otimes X$ equal $(X\otimes I)(I\otimes X)$? (By the parentheses I mean to signify normal matrix multiplication.)

$X$ is any unitary matrix of the same dimensions as $I$.

Answer 1

Yes. It doesn't even matter if $X$ is unitary. Use the definition of the tensor product of linear maps: if $f$ and $g$ are linear maps (i.e. matrices), and $v$ and $w$ are vectors, then

$$(f \otimes g)(v \otimes v) = f(v) \otimes g(w)$$

Now compose and check the identity you are suggesting:

\begin{align} (X \otimes I)(I \otimes X)(v \otimes w) &= (X \otimes I)(v \otimes X(w)) \\ &= X(v) \otimes X(w) \\ &= (X \otimes X)(v \otimes w) \end{align}

Answer 2

Using block matrix notation, $X\otimes X=[x_{i,j}X]_{i,j}$. $X\otimes I=[x_{i,j}I]_{i,j}$ and $I\otimes X=[\delta_{i,j}X]_{i,j},$ where $\delta_{i,j}$ is the Kronecker delta. Then the product of the latter two matrices may be computed using block matrix multiplication, giving: $$((X\otimes I)(I\otimes X))_{i,j}=\sum_{k}(x_{i,k}I)(\delta_{k,j}X)=x_{i,j}IX=x_{i,j}X=(X\otimes X)_{i,j}.$$