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?
[Math] Matrix Mathematical Notation
linear algebramatricesnotation
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In my own work, I use $\hat A$ to denote the column normalized form. This is to align with the use of hat (or circumflex) to denote normalized vectors: vectors denoted $\mathbf{u}$ and $\mathbf{\hat u}$ are often the unnormalized and normalized forms, respectively. Say $A \in \mathbb{R}^{n \times m}$, write the $A$ and $\hat A$ in terms of their column vectors
$$ A = \begin{bmatrix} \mathbf{a}_1 \ \ \mathbf{a}_2 \ \ \dots \ \ \mathbf{a}_m \end{bmatrix} \\ \hat A = \begin{bmatrix} \mathbf{\hat a}_1 \ \ \mathbf{\hat a}_2 \ \ \dots \ \ \mathbf{\hat a}_m \end{bmatrix} $$
where each column has been normalized as
$$ \mathbf{\hat a}_i = \frac{\mathbf{a}_i}{||\mathbf{a}_i||} $$
Note that this can be written as a decomposition. Scalar $k_i$ and column vector $\mathbf{k}$ are used for simplicity of notation.
$$ k_i := ||\mathbf{a}_i|| $$ $$ \mathbf{k} := \begin{bmatrix} k_1 \\ k_2 \\ \vdots \\ k_m \end{bmatrix} $$ $$ D_\mathbf{k} := \text{diag}(\mathbf{k})= \begin{bmatrix} k_1 &0 &&\dots &&&0 \\ 0 &k_2 &&\dots &&&0 \\ \vdots &\vdots &&\vdots &&&\vdots \\ 0 &0 &&\dots &&&k_m \\ \end{bmatrix} $$
Now we can write $A$ as
$$ A = \hat A D_\mathbf{k} $$
When I need to refer to the row normalized form and column normalized form separately, I just define and use $\hat R$ and $\hat C$, respectively. I've tried messing around with an arrow based notation to indicate direction of normalization, like $\hat A^\downarrow$ or $\underset{^\rightarrow}{A}$, but it's not worth the hassle and makes for noisy expressions.
The latter two notations are the same if one takes $\langle a,b \rangle :=\begin{pmatrix} a\\ b \end{pmatrix}$. This notation is sometimes used to have vectors written in-line. In this case, what you have is a linear combination of the two standard vectors:
$$x\langle 1,0 \rangle+y\langle 0,1 \rangle =x\begin{pmatrix} 1\\ 0 \end{pmatrix} + y\begin{pmatrix} 0\\ 1 \end{pmatrix} = \begin{pmatrix} x\\ y \end{pmatrix} = \langle x,y \rangle$$ Notice that these are both notations for vectors. The first notation is for a 2-by-2 matrix, which is not a vector unlike the latter two expressions. You can write the latter two expressions using the matrix-vector product, which might be the source of your confusion. In the 2-by-2 case the matrix-vector product is defined $$\begin{bmatrix} a &b \\ c & d \end{bmatrix}\begin{pmatrix} x\\ y \end{pmatrix} =x\begin{pmatrix} a\\ c \end{pmatrix}+y\begin{pmatrix} b\\ d \end{pmatrix}$$ From this it should be trivial to see what 2-by-2 matrix is used to give you your latter two expressions.
Best Answer
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.