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.
Best Answer
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.