10
votes

Accepted

### $3\times 3$ magic squares consisting of entries of a dense set $D\subseteq \mathbb{N}$

Yes.
By Szemerédi's theorem, your set contains an arithmetic progression of arbitrary length. In particular, it contains a progression of length 9, say it's $d_1,\ldots,d_9$. Then
$$
\begin{pmatrix}
...

6
votes

Accepted

### Solution to a Sylvester equation with positive definite coefficients

This is not true. For example, if
$$
A = \begin{bmatrix}
1 & 0 \\ 0 & 4
\end{bmatrix}, B = \begin{bmatrix}
4 & 0 \\ 0 & 1
\end{bmatrix}, C = \begin{bmatrix}
17 & 16 \\ 16 & 17
\...

5
votes

Accepted

### Eigenvalues invariant under 90° rotation

Assume $N$ is even (this is false when N is odd).
Let $X=2B, Y=A+A^T$.
Let
$$P = \begin{bmatrix}
1 & 1 & 1 & \cdots & 1 \\
1 & \zeta & \zeta^2 & \cdots & \zeta^{...

3
votes

### Full-rank matrix

OK, let's call the block matrix above $M$. First eliminate $N$ by a substitution $c\mapsto d N$. Then substitute $z_i \mapsto d y_i$ to eliminate $d$. Then you can construct the Schur complement w.r.t....

2
votes

### Non-singular matrix with restricted entries

[EXPANDED]
PART 1 (also done by Peter)
If $x,y$ are coprime and have opposite sign, there is a singular symmetric matrix with 1 on the diagonal and
only $x$ and $y$ off the diagonal.
Say $x<0,y>...

2
votes

### Non-singular matrix with restricted entries

Disclaimer: this is only a partial answer.
If $S = \{x, y\}$ (considered as variables), the determinant must be a polynomial with integer coefficients and constant coefficient 1. Therefore by Gauss's ...

1
vote

Accepted

### Spectral majorization for symmetric matrices

Yes. This notion of majorization of Hermitian matrices has been investigated before in the context of quantum information theory, and there are several good characterizations of this notion of ...

1
vote

Accepted

### A combinatorial matrix reconstruction problem

The meaning of the question is unclear due to the word "set". As Carlo proposed in a comment, I'll take it that we have an unordered list of $n$ unordered lists of $n$ elements, and we want ...

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