Row-normalized and column-normalized matrix notation I'm searching for the mathematical, algebraic notations of a row-normalized and column-normalized matrix.
For example, let us consider the following matrix A:
$$
A = \begin{pmatrix}
2 & 7 \\
4 & 3
\end{pmatrix}
$$
What is the mathematical notation of its corresponding row-normalized matrix?
$$
\begin{pmatrix}
2/9 & 7/9 \\
4/7 & 3/7
\end{pmatrix}
$$
What is the mathematical notation of its corresponding column-normalized matrix?
$$
\begin{pmatrix}
2/6 & 7/10 \\
4/6 & 3/10
\end{pmatrix}
$$
Best regards.
 A: $
\def\o{{\tt1}}
\def\LR#1{\left(#1\right)}
\def\op#1{\operatorname{#1}}
\def\Diag#1{\op{Diag}(#1)}
\def\trace#1{\op{Tr}\LR{#1}}
\def\qiq{\quad\implies\quad}
\def\c#1{\color{red}{#1}}
\def\CLR#1{\c{\LR{#1}}}
$Let $(\oslash)$ denote elementwise division and $J$ the all-ones matrix, then
$$\eqalign{
B &= A\oslash AJ,\qquad  C &= A\oslash JA \\
}$$
where $B$ is row-normalized and $C$ is column-normalized.
If you prefer, the all-ones vector $\o$ and the Diag operator can be employed to replace elementwise division with the matrix inverse of a diagonal matrix
$$\eqalign{
B &= \Diag{A\o}^{-1}A,\qquad  C &= A\:\Diag{A^T\o}^{-1} \\
}$$
A: 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.
