# How to denote matrix concatenation?

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.

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 $$[\begin{array}{l}A& B\end{array}]$$.
• 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. Aug 16 '13 at 14:40
• 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: $\begin{bmatrix}A_{(k\times m)} & B_{(k\times n)} \end{bmatrix}$. Aug 19 '13 at 9:19
In the example given, $$C = \begin{bmatrix} A \\ B \end{bmatrix}$$ is a standard notation. This is a particular example of "block notation".