TITLE: Prove the Hadamard product representation
QUESTION [8 upvotes]: Let $A$ and $B$ be $m \times n$ matrices with low-rank structures:
$$
A = U_{A}\Sigma_{A}V_{A}^{T},\quad B= U_{B}\Sigma_{B}V_{B}^{T},
$$
Prove that Hadamard product $A\circ B$ admits the following representation
$$
A\circ B = (U_{A}^T\odot U_{B}^T)^T (\Sigma_{A}\otimes\Sigma_{B})(V_{A}^{T}\odot V_{B}^{T}),
$$
where $\odot$ represents the Khatri-Rao product, and $\otimes$ the Kronecker product.

REPLY [3 votes]: We will use the following properties and definitions:
\begin{align}
    (A\odot^T B)(C\odot D) = (AC)\circ(BD),\label{eq:khr-had}\\
    (A\otimes B)(C\odot D) = (AC)\odot(BD),\label{eq:khr-kro}
\end{align}
It is easy to prove that Hadamard product of A and B admits the following representation:
\begin{align}
    A \circ B 
    &= (U_{A}\Sigma_{A}V_{A}^{T})\circ(U_{B}\Sigma_{B}V_{B}^{T})\nonumber\\
    &= (U_{A}^T\odot U_{B}^T)^T (\Sigma_{A}V_{A}^{T}\odot \Sigma_{B}V_{B}^{T})\nonumber\\
    &=(U_{A}^T\odot U_{B}^T)^T (\Sigma_{A}\otimes\Sigma_{B})(V_{A}^{T}\odot V_{B}^{T})\label{eq:repres}
\end{align}
where $\odot$ represents the Khatri-Rao product, and $\otimes$ the Kronecker product.