I am interested in $n \times n$ matrices over some field $K$ all whose rows and all whose columns sum to zero.

First question: do these matrices have a name?

Pending an answer I will call these "null-matrices".

Second (main) question:

Given $n$, are there subsets $J \subset \{1, \ldots n\} \times \{1, \ldots, n\}$ of indices such that $$\sum_{(i, j) \in J} a_{i,j} = 0$$ for every null-matrix $(a_{ij})_{i, j = 1 \ldots n}$?

More visually: can you take a red pencil and put red circles around some entries in a an (still empty) matrix so that in whichever way someone fills up the matrix with elements of $K$ to obtain a null-matrix, the sum of the red-circled entries will always add up to zero?

Obviously the answer is yes, just take $J$ to be a disjoint union of rows or a disjoint union of columns. So my question is: are there examples of set $J$ that are not of this form?

In general (i.e. without specifying the field over which we consider the matrix) I expect the answer to be no (but please prove me wrong) - however for some special combinations of $n$ and char($K$) the answer might be yes. In particular, for $n = 2$, char($K$) $=2$, the diagonal (i.e. $J = \{(1, 1), (2,2)\}$) is an example of the type of set I'm looking for.

Are there more examples like this?

  • $\begingroup$ Could not seem to find a name for such a matrix on this list en.wikipedia.org/wiki/List_of_matrices, but does not mean it does not have a name. $\endgroup$ – Amzoti Dec 24 '13 at 14:26
  • $\begingroup$ I am not sure too which extent this might be helpful, but such matrices are precisely the matrices of the form $I+A$, where $A$ is a doubly stochastic matrix. $\endgroup$ – Martin Sleziak Dec 24 '13 at 15:17
  • 1
    $\begingroup$ @MartinSleziak: You mean $A-I$ (or $I-A$). $\endgroup$ – Robert Israel Dec 24 '13 at 16:06
  • $\begingroup$ Vincent, I too am interested in these matrices. I am also interested in those matrices whose rows sum to $0$, with no constraints on the columns. Both form an $S$-subalgebra of the $S$-algebra $\mathrm{Mat}_n(S)$, where $S$ is any semiring (with $0$ and $1$, but not necessarily commutative.) The former are of course also closed under transposition. Anyway, did you end up finding out anything interesting about them? $\endgroup$ – goblin Apr 17 '15 at 10:20

If $2=0$, examples can be constructed by partitioning the index set for the rows and columns into two subsets $A$ and $B$, and taking the subset $(A \times A) \cup (B \times B)$.

No nontrivial examples exist in characteristic $\neq 2$. Considering matrices that are $0$ outside a $2 \times 2$ minor shows that if a subset meets the conditions and contains two positions of the matrix that do not share a row or columns, it has to contain all $4$ corners of the $2 \times 2$ minor with those two points as a diagonal. Therefore, the subset is a Cartesian product of a subset of rows with a subset of columns. The only such sets with guaranteed zero sum of entries are unions of several complete rows, or of several complete columns.


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