What is the process of combining vectors into a matrix called?
My question is not how or why to do this. My question is, "What is this process called?"
For example, if we need a solution for a system of linear equations, one method is to represent the unknowns as column vectors and then join those column vectors into a single matrix. One way to describe it might be to say "indexing" an array but I'm looking for a formal terminology. Ultimately, I'd like to know how to represent this in a generalized form.
Essentially, I'm asking how to describe and formally write something like this:
$$
\begin{equation}
\vec v_1=\left(\begin{array}{c} X_1 \\ Y_1 \\ Z_1 \end{array}\right), \vec v_2=\left(\begin{array}{c} X_2 \\ Y_2 \\ Z_2 \end{array}\right) \rightarrow M=\left(\begin{array}{cc} X_1 & X_2 \\ Y_1 & Y_2 \\ Z_1 & Z_2 \end{array}\right)
\end{equation}
$$
Thanks.
Best Answer
We call a matrix produced by concatenation of two appropriately sized matrices (not necessarily vectors) an augmented matrix. To emphasize that we formed such a matrix in this way, we might write (continuing the given example) $$M = \left(\begin{array}{c|c}{\bf v}_1 & {\bf v}_2\end{array}\right) = \left(\begin{array}{c|c}X_1 & X_2 \\ Y_1 & Y_2 \\ Z_1 & Z_2 \\ \end{array}\right) .$$ Depending on context, I might not use this term when concatenating more than two vectors, that is, when forming for vectors ${\bf w}_1 , \ldots, {\bf w}_r$ the matrix $$\left(\begin{array}{ccc}{\bf w}_1 & \cdots & {\bf w}_r\end{array}\right) .$$ (In this case I would generally omit the bars, $\vert$.)