# Prove the Hadamard product representation

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.

• The Khatri-Rao product, for those unfamiliar Commented May 26, 2019 at 16:26
• This is an interesting problem, where did you encounter it? Also, could you clarify how exactly $U_A$ is meant to be partitioned for the Khatri-Rao product? Commented May 26, 2019 at 16:29
• @Omnomnomnom $U_{a}$ is the matrix of left singular vectors of A. Commented May 26, 2019 at 16:43
• Commented May 26, 2019 at 18:54

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.