At most one positive entry in each column: How to write this in a mathematical way?

Question

Let $\mathbf{X} =(x_{ij}) \in \mathbb{R}^{I \times J}$ a matrix and $\mathbf{x}_j$ be its $j$th column vector.

Statement : There is at most one positive entry in each column vector of $\mathbf{X}$ and 0 otherwise. If there is a positive entry, denote the index of that entry as $I(j)$ (or any notation is possible).

How can I represent this statement with mathematical notation?

This was my trial. \begin{equation} H(x) = \begin{cases} \displaystyle \infty, & x < 0 \\ 0,& x = 0\\ 1, & x > 0 \end{cases} \end{equation} \begin{equation} SH(\mathbf{x}_j) = {\sum_{i=1}^{I} H(x_{ij})} \end{equation}

Statement : $SH(\mathbf{x}_j) \leq 1, \forall j, 1 \leq j \leq J$

I'm not sure about the infinite part, and I think there is a better notation. As I'm new in mathematical writing, any ideas or tips would be appreciated.

Answer 1

Welcome to MSE! I'm not 100% sure I understand your question, but I can think of two interpretations.

(1) If you're looking to translate the property into a mathematical formula, you can say $$\forall j\in\{1,\ldots,J\},\;\forall i\in\{1,\ldots,I\},\;x_{ij}\geq 0,$$ and $$\forall j\in\{1,\ldots,J\},\;\#\{i\in\{1,\ldots,I\}\mid x_{ij}>0\}=1.$$

(2) If you're looking for a function $f$ such that $f(\mathbf{X})=0$ iff it satisfies the property, then consider that $\mathbf X$ satisfies the property iff $\forall i,j,\;x_{ij}\geq 0$ and $$\forall j\in\{1,\ldots,J\},\;\sqrt{\sum_{i=1}^Ix_i^2}=\sum_{i=1}^Ix_i,$$ so we can set $$f(\mathbf X):=\left(\sqrt{\sum_{i=1}^Ix_i^2}-\sum_{i=1}^Ix_i\right)^2+\left(\sum_{i,j}|x_{ij}|-x_{ij}\right)^2.$$ Okay, it's not very pretty, but it gets the job done :)

Answer 2

Your formulation is correct (as noted in the comments you can take $H(x)=2$ rather than $\infty$ when $x<0$), however, as you have probably noticed it is extremely complicated and hard to understand, essentially requiring the reader to decipher what you mean. You should almost certainly just write what you mean in English. The plain-english formulation in your question is a little hard to understand, but this is more a matter of slightly sloppy writing than of not having written everything in mathematical formulae. You could write:

The matrix $X$ has non-negative entries with at most one strictly positive entry per column. If the $i$-th column contains a strictly positive entry $x$, denote the row-index of that value by $I(i)$.