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Let ${\bf G} \in\mathbb{C}^{M\times K}$ and ${\bf H} \in\mathbb{C}^{N\times K}$ are full-rank Vandermode matrices where $MN-1=K>N\geq N$, that is, ${\bf G}$ and ${\bf H}$ are fat. Let ${\bf F}= {\bf H}\circ {\bf G}$ where $\circ$ denotes Khatri-Rao matrix product or column-wise Khatri-Rao matrix product.

Is it true that there exists ${\bf g} \in\mathbb{C}^{M}$ and ${\bf h} \in\mathbb{C}^{N}$, which are different from the columns of ${\bf G}$ and ${\bf H}$, such that ${\bf h}\otimes{\bf g}$ belongs the range space of $\bf F$, that is,

\begin{equation} {\bf F}{\bf a} = {\bf h}\otimes{\bf g}? \end{equation}

Thanks.

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I assume that $\otimes$ is the Kronecker product? –  Calle Jan 19 at 17:45

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