$\DeclareMathOperator{\rk}{rk}$


The question below is implicit in [this MO post](https://mathoverflow.net/questions/265468/pointwise-hadamard-matrix-product-and-the-rank), but I believe it deserves to be asked explicitly, particularly now that I have some more numerical evidence.



> 
> Suppose that $A$ is a real, square matrix of order $n$, such that all main-diagonal elements of $A$ are distinct from $0$, and of any two elements symmetric about the main diagonal, at least one is equal to $0$. That is, writing $A=(a\_{ij})\_{1\le i,j\le n}$, we have $a\_{ii}\ne 0$ and $a\_{ij}a\_{ji}=0$ whenever $i,j\in[1,n]$, $i\ne j$. In other words, $A$ can be obtained from a non-singular triangular matrix by switching some pairs of symmetric elements. How small can the rank of such a matrix be, in terms of $n$?
> 
> 
> 


Since the pointwise product $A\circ A^t$ is full-rank, as an immediate corollary of [the inequality](https://en.wikipedia.org/wiki/Hadamard_product_(matrices)) $\rk(B\circ C)\le\rk(B)\,\rk(C)$ we have $\rk(A)\ge\sqrt n$. How sharp this estimate is? Is it true that $\liminf\_{n\to\infty} \log(\rk(A))/\log(n)=\frac12$?


Computations show that there are matrices of order $6$ and rank $3$ satisfying the assumptions above; taking $A$ to be a tensor power of such a matrix, we will have $\rk(A)=n^c$ with $=\log\_6(3)\approx0.6131$.


I do not know whether there exist matrices of order $7$ and rank $3$ with the property in question.



The rank of a perturbed triangular matrix
$\DeclareMathOperator{\rk}{rk}$It is possible to construct a matrix with $\rk(A)\leq 2\sqrt{n}$. Assuming that $n=r^2$ with an integer $r$, introduce two matrices $B$ and $C$, whose rows and columns are indexed by elements of $\{1,2,\dotsc,r\}^2$, and whose entries are defined by
$$
B\_{(x,y),(x',y')}=\begin{cases}1&\text{if }x=x',\\0&\text{otherwise},\end{cases}
$$
and
$$
C\_{(x,y),(x',y')}=\begin{cases}-1&\text{if }y<y',\\0&\text{otherwise},\end{cases}
$$
where $(x,y),(x',y')\in \{1,2,\dotsc,r\}^2$.


We have $\rk(B)\leq r$ and $\rk(C)\leq r$, and $A:=B+C$ is an $r^2$-by-$r^2$ matrix satisfying the requisite condition.


This is a minor modification of the constructions that I used in a [paper of mine](http://arxiv.org/abs/1508.00145).


