146 problems

For integers $n\geq 6$, let $D_{n,3}$ denote the matrix considered in the paper, and let ${\mathop{\mathrm{rank}}}_{\mathrm{bin}}$ denote binary rank. Hefner–Henson–Lundgren–Maybee…

Let $A$ and $B$ be square matrices of the same size. Their permanents are multiplicative when … Marcus–Minc conjecture. Multiplicativity holds only when both $A$ and $B$ are produc…

Pate's conjecture. The largest eigenvalue of $\mathscr{C}_k(A)$ is $\operatorname{per}(A)$. This conjecture is refuted; the paper states that its $k=1$ case has an $8\times8$ count…

Let $H$) be an $n\times n$ positive semi-definite Hermitian matrix, and let $\pi(H)$ denote its Schur power matrix, whose entries are indexed by permutations in $S_n$. Soules's per…

Let $\mathbb{F}$ be a field and let $\operatorname{Mat}_n(\mathbb{F})$ be the algebra of $n\times n$ matrices over $\mathbb{F}$. Let $\Gamma(\mathbb{F},n)$ be the graph whose verti…

Bourque–Ligh conjecture. The LCM matrix $[S]$ is invertible for every such gcd-closed set $S$.

Kontsevich's Hadamard-inverse rank conjecture. The rank of $Hinv(A)$ is never two, although every other rank might be possible. This is a concrete problem about rank constraints fo…

Conjecture 4. Every such row is one of

Let $A$ and $B$ be two hypomorphic matrices, meaning that $B=\Sigma(A)$ for a hypomorphism $\Sigma$ as in the surrounding reconstruction setting. Let $\lambda_i$ be a simple eigenv…

Let $S_n$ be the permutation group on $n$ letters and let $\mathbb Z_2^n$ denote the group of diagonal sign matrices. Twisted-product refinement. The group $G(A)$ in the spectral r…

Let $A$ be a real $n \times n$ symmetric matrix. For each $i$, write $A_i$ for the matrix obtained by deleting the $i$th row and column, and let $\operatorname{eigen}(A)$ denote th…

The Asymptotic Lower $r$-Permanent Conjecture. Under these hypotheses,

Varga's conjecture. Every nonsingular tau-matrix satisfies this inequality. The bound was known in dimensions $n\leq 4$ at the time described, but the paper gives unstable GKK tau-…

For a matrix $A$, let $l(A)$ be the least real eigenvalue of $A$ when $A$ has a real eigenvalue, and let $l(A)=\infty$ otherwise. An omega-matrix has eigenvalue monotonicity … A ta…

Let $D=(V,A)$ be a digraph. For distinct vertices $x,y\in V$, write $x\rightarrow y$ when $xy\in A$, and let $N^+(x)$ and $N^-(x)$ denote the sets of out-neighbors and in-neighbors…

Extended local-permutation conjecture. Every element of $E(\mathfrak{t}_{\mathrm{loc}})$ is similar to a block-shift matrix via an element of $\Pi_{\mathrm{loc}}^{\mathrm{ex}}$.

M-theory is described, at least in the infinite momentum frame, by the dimensional reduction to one dimension of ten-dimensional supersymmetric Yang--Mills theory, denoted…

A q-graded Markovian matrix is intended to generalize the bi-graded Markovian matrices discussed above, although the source does not define the grading or the required Markovian pr…

Symmetrized trace conjecture. For matrices of higher degree, the trace of $Q$ is given by the symmetrized-trace expression proposed in the continuation of the source passage.

Kontsevich's conjecture. The rank of $B$ is never equal to three.

Let $S_n$ be the symmetric group on \\{1,\ldots,n\}. For an $n\times n$ matrix $A=(a_{ij})$, define … Let $K_n$ be the set of all real nonnegative $n\times n$ matrices whose entr…

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…

Degree-bound conjecture. The algebra $\mathcal E_n\langle X,Y\rangle$ should be spanned, as a module over the ring of invariants, by monomials of degree

Determinant and inverse conjecture. One has