I ran across the face-splitting product recently, but there is a frequently mentioned property that I cannot make sense of. The property is Property 5 at Wiki's Khatri-Rao page: $$\mathbf{A} \otimes (\mathbf{B} \bullet \mathbf{C}) = (\mathbf{A} \otimes \mathbf{B}) \bullet \mathbf{C}$$ where $\bullet$ is the face-splitting version of the Khatri-Rao product and $\otimes$ is Kronecker product. This property is also frequent in Slyusar's papers, but so far I have failed to locate the original reference for it.
Now, in $\mathbf{B} \bullet \mathbf{C}$, the number of rows in $\mathbf{B}$ and $\mathbf{C}$ must be the same for $\mathbf{B} \bullet \mathbf{C}$ to be defined. But the number of rows in $\mathbf{C}$ must equal the number of rows in $\mathbf{A} \otimes \mathbf{B}$ for the right hand side to be defined. This implies that $\mathbf{A}$ is a row vector. With $\mathbf{A}$ a row vector, the property is easily verified to be correct.
However, all notation I have come across, both Wiki and Slyusar's papers, strongly indicate that $\mathbf{A}$ is a matrix. The reason is that there are multiple properties only true for vector-valued quantities, and then the notation $\mathbf{a}$ is used.
P.S. I would not ask this question if I had only been confused from Wiki. I mention Wiki since it is the most accessible source for anyone interested in this post. Wiki's list of properties is essentially taken from Slyusar's papers on FSP.
Thanks/Fredrik