Matrix Mathematical Notation

Question

I am trying to work out the mathematical notation for combining the columns of two matrices, $$A=\begin{pmatrix}1 & 3 & 5 \\ 2 & 4 & 1 \\ 3 & 7 & 9\end{pmatrix}$$ and $$B=\begin{pmatrix}4 & 4 & 3 \\ 9 & 10 & 11 \\ 12 & 15 & 13\end{pmatrix},$$ to form the new matrix $$C=\begin{pmatrix}1 & 4 & 3\\ 2 & 10 & 11 \\ 3 & 15 & 13\end{pmatrix}.$$ $C$ is a matrix which is made up of the first column of $A$ and the last two columns of $B$. The problem I have is expressing $C$ in terms of $A$ and $B$ using appropriate mathematical notation, I can code it, I just don't know the notation for it! Any suggestions?

Answer 1

Let $E_{ij}$ be a matrix where all it's entries are $0$ except the entry located on the $i^{th}$ row and $j^{th}$ column which is equal to $1$. Then we have $$C=AE_{1,1}+B(E_{2,2}+E_{3,3})$$

Answer 2

Note that $C=\underbrace{A\begin{bmatrix}1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{bmatrix}}_{\text{Saves first column of }A}+\underbrace{B\begin{bmatrix}0 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}}_{\text{Saves last two columns of }B}$.

You should be able to generalize.

Answer 3

There's no standard notation for such an operation. You can use whichever you feel appropriate. I'd try to describe it as $$A=\left[a_1,a_2,a_3\right],\quad B=\left[b_1,b_2,b_3\right],$$i.e. a matrix is a set of columns seen as vectors, then $$C=\left[a_1,b_2,b_3\right].$$

Answer 4

Describing the columns of the matrices as vectors, you can also say $$ \begin{align*} A &= \begin{bmatrix}\mathbf{a}_1 & \mathbf{a}_2 & \mathbf{a}_3 \end{bmatrix}, \\ B &= \begin{bmatrix}\mathbf{b}_1 & \mathbf{b}_2 & \mathbf{b}_3 \end{bmatrix}, \\ C &= \begin{bmatrix}\mathbf{a}_1 & \mathbf{b}_2 & \mathbf{b}_3 \end{bmatrix}. \end{align*} $$ where the reader will instantly see what is happening. But of course this is not a "mathematical operation" to obtain $C$ from $A$ and $B$ as in the other answers.