# How to determine the rank of a Khatri-Rao product of two matrices based on their each rank

As is known to all, the Khatri-Rao product is defined as $$\mathbf{C}=\mathbf{A}\odot \mathbf{B}=\left[\begin{matrix}\mathbf{a}_1\otimes\mathbf{b}_1&\mathbf{a}_2\otimes\mathbf{b}_2&\cdots \mathbf{a}_K\otimes\mathbf{b}_K\end{matrix}\right]$$,where both $$\mathbf{a}_i\in \mathbb{C}^{I\times1}$$ $$\mathbf{b}_i\in \mathbb{C}^{J\times1}$$ and $$\mathbf{C} \in \mathbb{C}^{IJ\times K}$$. I don't know how to determine the rank of the matrix $$\mathbf{C}$$,based on the ranks of $$\mathbf{A}$$ and $$\mathbf{B}$$.

• Any references could be found on it? – DuWeimin Nov 12 '18 at 3:22