30 problems

Let $A$ be a square matrix over a subring $\mathcal R$ of $\mathbb R$. Its nonzero spectrum is the multiset of its nonzero eigenvalues. Boyle–Handelman's strong generalized spectra…

Entrywise nonnegativity conjecture. This equality actually implies that

Let $\sigma=\{a,b,c\pm di\}$ satisfy the preceding case assumptions: $\{a,c\pm di\}$ is $\mathcal{R}$, $a+2c>0$, $a-c-\sqrt{3}d=0$, and $b>\frac{a+2c}{3}$. Thus…

Let $\sigma$ be a spectrum, equivalently the spectrum of a polynomial $P(x)$. Write $\mathcal{R}$ for realizability, $\mathcal{IR}$ for irreducible realizability, and SS for the st…

Let $A$ be an $n\times n$ non-negative, non-singular matrix. An S-matrix is a matrix in $\{0,1\}^{n\times n}$ satisfying … where $\boldsymbol{1}$ is the all-ones vector. Harwitz–Sl…

Reality-and-inequalities conjecture. The set of spectra realizable by nonnegative matrices is solvable with the reality condition and $\lfloor (n+1)/2\rfloor$ unions of $n$ polynom…

Basic semialgebraicity conjecture. The set of spectra realizable by symmetric nonnegative matrices forms a basic semi-algebraic set.

Let $\mathcal{P}_{n,2n}$ be the cone of polynomials of degree at most $2n$ that preserve nonnegative matrices of order $n$, and let $\mathbf{v}$ denote the coefficient vector of su…

Normalized-volume decay conjecture.

Continuity conjecture. The largest $t\in\mathbb{R}^+$ such that $p_t\in\mathcal{P}_n$ for all such $m,s$ is a continuous function of $m$ and $s$, bounded between $1$ and $2$. The c…

Let $\mathcal{P}_{n,m}$ be the cone of degree-at-most-$m$ polynomials that preserve nonnegative matrices of order $n$, and consider its volume after restriction to the unit sphere.…

Let $\mathscr{S}(n,e)$ be the set of $n\times n$ $(0,1)$-matrices with exactly $e=c^2+t$ ones, where $2\leq t\leq 2c$. If $A\in\mathscr{S}(n,e)$ attains the maximum spectral radius…

Let $A$ be a nonnegative substochastic matrix, meaning … for all $i$ and $j$. Suppose that above the diagonal, $A$ has nonzero entries only at distance $1$ from the diagonal, while…

Consider the Markov chain on $\mathbb{N}$ in which every vertex $i$ has an edge of weight $r$ to $i-1$ and edges of weight $c_l$ to $i+l$ for $0\leq l\leq k$, with … Let $p_l$ be t…

Let $A$ be a nonnegative substochastic matrix, meaning … for all $i$ and $j$. Suppose that above the diagonal, $A$ has nonzero entries only at distance $1$ from the diagonal, while…

Let $A$ be a nonnegative substochastic matrix, meaning … for all $i$ and $j$. Suppose that above the diagonal, $A$ has nonzero entries only at distance $1$ from the diagonal, while…

Let $C_n$ denote the $n\times n$ Chet matrix, and let $C=\{n:C_n\text{ is entry-wise nonnegative}\}$. Chet Conjecture. One has … More strongly, … The paper presents this as its mai…

Let $n$ be a positive integer, and let … Here $p(A)$ is the polynomial evaluated at the matrix $A$, and the inequality is entrywise. Clark–Paparella conjecture. For every positive…

For $n\geq 3$, let $\mathscr{P}_n$ denote the set of polynomials that preserve entrywise nonnegative $n\times n$ matrices under polynomial evaluation. Hierarchy conjecture. … The p…

Let $\mathcal R$ be a subring of $\mathbb R$, and let $\Lambda=(\lambda_1,\dots,\lambda_k)$ be a $k$-tuple of complex numbers. The necessary conditions are the Perron condition, th…

Let $N$ be a non-negative real and $k$ a positive integer. For a square matrix $A$ with non-negative real entries, define its weight by … Assume that $A$ is nilpotent and has weigh…

Let $N$ be a non-negative real and $k$ a positive integer. For a square matrix $A$ with entries in ${\mathbb R}_{\ge 0}$, define its weight by … Assume that $A$ is nilpotent and ha…

Aligning spectral-radius characterization conjecture. The right-hand term in Proposition 3.1 does, in fact, characterize $\rho^a(A)$.

Asarin's conjecture. The equality above should be valid for certain classes of non-negative matrices.

Let $CE_n$ denote the critical exponent for $n$-by-$n$ generalized doubly nonnegative matrices, and let $MIP_n$ be the maximum index of primitivity among primitive $n$-by-$n$ gener…