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How do you call it when you stack two vectors, let's say $u=\pmatrix{u_1\\u_2}$, $v=\pmatrix{v_1\\v_2}$, on top of each other such that you get $$u\oplus v=\pmatrix{u_1\\u_2\\v_1\\v_2}?$$ I found this notation in this paper, chapter II.A. From what I could find I suspect it is either a direct sum or a Kronecker sum but I don't know how those sums apply to vectors, only matrices.

AccidentalTaylorExpansion
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    "Concatenation" is what I'd call it. There isn't necessarily a symbol used for this operation and it doesn't necessarily appear outside of very specific applications, so if you have reason to use this operation just be sure you state what it is what you are doing. – JMoravitz Jun 20 '22 at 19:41
  • @JMoravitz That's weird, I would have thought that this operation would have been more common. Thanks for the clarification! – AccidentalTaylorExpansion Jun 20 '22 at 20:01
  • I call this “concatenation” or perhaps “vertical concatenation”. – littleO Jun 20 '22 at 20:24
  • @AccidentalTaylorExpansion To address your thought that the operation would be more common: generally speaking, when we combine two objects of the same "type", we might hope for a third object of the same "type". This doesn't always happen (inner products, for example, combine vectors to obtain scalars), but it is a "desirable" property when defining an operation. This concatenation of vectors takes two vectors of dimensions $m$ and $n$, and produces a vector of dimension $m+n$. This is kind of a weird thing to do (keeping in mind that vectors are a bit more than just ordered lists). – Xander Henderson Jun 20 '22 at 20:32
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    That being said, I do remember attending a research seminar in which this kind of operation did come up. It is not my area of expertise, but I believe that the example had something to do with von Neumann algebras (and the factors thereof). As I said, I am not an expert, but the trick was to embed matrix algebras into larger spaces via concatenation, e.g. by forming block matrices of the form $$\begin{pmatrix} A & 0 \ 0 & B \end{pmatrix}, $$ where $A$ and $B$ are square matrices (of, perhaps, differing dimension). But in that context, I believe that this reduces to a tensor product... – Xander Henderson Jun 20 '22 at 20:44
  • ...so it has a kind of different flavor than just straight up concotenation. – Xander Henderson Jun 20 '22 at 20:45
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    @XanderHenderson Yeah, that's very common in representation theory -- if you have two representations of a group $G$ such that $\Pi_1: G\rightarrow \mathrm{GL}(V_1)$ and $\Pi_2:G\rightarrow \mathrm{GL}(V_2)$, you can form a new representation $\Pi \rightarrow \mathrm{GL}(V_1 \oplus V_2)$ where $\forall g\in G,\Pi(g)=\begin{pmatrix}\Pi_1(g)&0\ 0&\Pi_2(g)\end{pmatrix}$. In that case, you can form elements $v \in V_1\oplus V_2$ by taking $v_1 \in V_1, v_2\in V_2$ and writing $v=\begin{pmatrix}v_1\v_2\end{pmatrix}$ – kc9jud Jun 24 '22 at 17:45

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Looking at Notation for set of concatenated vectors?, as @JMoravitz says, "concatenation" is probably the best name for it. Notationally, it's quite common to write this as $$w = \pmatrix{u\\v}$$ where $u \in U, v \in V, w\in W \cong U \oplus V $.

This is basically the same notation as block matrices, where the "matrix" in question is $n \times 1$.

kc9jud
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