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Is there a name for combining two vectors $\vec{x} = \begin{bmatrix} x_0 \\ x_1 \\ \vdots \\ x_n \end{bmatrix}$ and $\vec{y} = \begin{bmatrix} y_0 \\ y_1 \\ \vdots \\ y_n \end{bmatrix}$ into a matrix

$$\begin{bmatrix}x_0 & y_0\\x_0 & y_1\\ \vdots & \vdots \\x_0 & y_m\\x_1 & y_0\\x_1 & y_1\\ \vdots & \vdots \\x_1 & y_m\\ \vdots & \vdots \\x_n & y_0 \\ x_n & y_1 \\ \vdots & \vdots \\ x_n & y_m\end{bmatrix} ?$$

How is this operation or this result called?

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It's a block matrix $\begin{bmatrix}X& Y\end{bmatrix}$ of two columns $X=\vec x\otimes \vec 1_m$ and $Y=\vec1_n \otimes \vec y$, where $\vec 1$ is a vector of all ones, $\otimes$ is the Kronecker product.

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I believe the phrase you want is 'augmented matrix'.

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  • $\begingroup$ Looking here, the augmented matrix is not what I'm doing. I'm not just pairing elements with the same indices, but the target matrix has a row number of $|\vec{x}| \cdot |\vec{y}|$, so all element combinations appear as rows. $\endgroup$ – Emit Taste Oct 25 '16 at 17:57

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