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hope this question isn't too trivial.

So, I have a vector $v\in\mathbb{R}^{1\times k}$ and the following Kronecker product: $$M = \mathbb{I}_{k} \otimes v = \begin{bmatrix} v &0 &\dots &0 \\ 0 & v & \dots &0 \\ \vdots & & \ddots &\vdots \\ 0 & & \dots& v\end{bmatrix} \in\mathbb{R}^{k\times k^2}$$

Is there any simple way to write $M$ as a regular matrix multiplication? Say $M = T\,v\,G$, with T and G of appropriate size. I need the vector $v$ outside the Kronecker product.

Thanks!!!

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Certainly, it's not possible to get $M = TvG$ for any $T$ and $G$. Notably, $M$ has rank $k$, but $v$ has rank $1$, which means that the rank of $TvG$ is at most $1$.

I can't think of a form which is more convenient than that Kronecker product.

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  • $\begingroup$ Thanks for the early reply. The thing is that I need the vector $v$ explicitly outside the Kronecker product. Would you say that there is no combination i.e. $T\,v\,G$ that can lead me to $M$? If that's the case, I'll probably have to rewrite my problem in a more suitable form. $\endgroup$ – Nico F. Sep 1 '17 at 9:52
  • $\begingroup$ That's exactly what I said $\endgroup$ – Omnomnomnom Sep 1 '17 at 11:06

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