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Trivial question: Is there any standard notation for the concatenation of two or more matrices?

Example:

$$A = \left(\begin{array}[c c] - a_1 & a_2\\ a_3 & a_4 \end{array}\right),$$

$$B = \left(\begin{array}[c c] - b_1 & b_2\\ b_3 & b_4 \end{array}\right),$$

Then the concatenation (by rows) of $A$ and $B$ is:

$$C = \left(\begin{array}[c c] - a_1 & a_2\\ a_3 & a_4 \\ b_1 & b_2\\ b_3 & b_4 \end{array}\right).$$

I just want to know if there is a standard notation for this operation.

Anibal Troilo
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2 Answers2

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There is such a thing as "augmenting" two matrices. For example, augmenting your matrices $A$ and $B$ above gives $$(A\mid B)=\left(\begin{array}{ll}a_1&a_2\\a_3&a_4\end{array}\left|\begin{array}{ll}b_1&b_2\\b_3&b_4\end{array}\right.\right).$$ This is useful notation for Gaussian row reduction since it makes clear the two matrices.

You can also write $[\begin{array}{l}A& B\end{array}]$.

pshmath0
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  • Thanks. Is this usually denoted as $(A\vert B)$? In my case the dimensions of the matrices of interest are of the order of hundreds. – Anibal Troilo Aug 16 '13 at 14:40
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    Yes, usually you would write the augmentation as $(A|B)$ or simply as a block matrix $\begin{bmatrix}A & B \end{bmatrix}$. If you want to indicate the dimensionality, you could also do: $\begin{bmatrix}A_{(k\times m)} & B_{(k\times n)} \end{bmatrix}$. – B0rk4 Aug 19 '13 at 09:19
  • Answer updated with comments in mind. – pshmath0 Jul 08 '20 at 12:55
  • I find it interesting the way you seem to make a distinction between square and round brackets for matrix notation, is that distinction spelt out somewhere in a text book? It is not something that's standard I think. – Joce Apr 09 '21 at 12:56
  • I've seen both styles used; I guess it may depend on the field. – pshmath0 Jan 17 '22 at 17:21
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In the example given, $$ C = \begin{bmatrix} A \\ B \end{bmatrix} $$ is a standard notation. This is a particular example of "block notation".

littleO
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