# 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.

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.