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For example, I want to this to happen: $$\begin{bmatrix}1& 2& 3\end{bmatrix}\times\begin{bmatrix}2& 3& 4\end{bmatrix} = \begin{bmatrix}2& 6& 12\end{bmatrix}$$ It's not exactly matrix multiplication, but I hope you can see what I'm getting at. Is there some notation in linear algebra that allows this function to be valid?

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I believe what you are describing is called the Hadamard product. You can read about them and see some notation for using them on the Wikipedia page. https://en.wikipedia.org/wiki/Hadamard_product_(matrices)

This page uses the notation $\circ$ for the Hadamard product.

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  • $\begingroup$ That is what I know it as. $\endgroup$ – Carser Aug 15 '16 at 12:55
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I don't know any standard matrix operation of this kind. You can rewrite the three entries of the $1\times 3$ matrices as diagonal elements of a $3 \times 3$ matrix, than perform the standard matrix multiplication.

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