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.
$ \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
Diagoperator can be employed to replace elementwise division with the matrix inverse of a diagonal matrix $$\eqalign{ B &= \Diag{A\o}^{-1}A,\qquad C &= A\:\Diag{A^T\o}^{-1} \\ }$$