15 problems

Let $S(X,B)=P$ be a symmetric word equation in positive definite matrices, with $X$, $B$, and $P$ of size $2\times2$. Two-by-two unique-solvability conjecture. Symmetric word equat…

Let $S(X,B_i)=P$ be a symmetric word equation, where $X$ is an $n\times n$ positive semidefinite matrix, the $B_i$ are fixed $n\times n$ positive definite matrices, and $P$ is an…

Nonexistence conjecture. There exist no $(p+1)$-tuples of matrices $A_j$ or $M_j$ satisfying the additive or multiplicative matrix equations, respectively, with trivial centralizer…

Deligne–Simpson conjecture. Conditions i) and ii) from Theorem are necessary and sufficient if $q=1$. For $q>1$, there are cases in which they are sufficient and cases in which the…

A left-unilateral matrix equation is an equation … where the coefficients $a_r$ and the unknown $x$ are square matrices of the same order, with all coefficients on the left. An ite…

Smallest-root conjecture. $\theta$ must be the smallest positive root of the displayed equation. The claim selects the physically relevant positive root in the exact-relations cons…

Nilpotence conjecture. One has

Let $B$, $C$, and $E$ be real matrices satisfying … The zero entries of $E$ are contained in a submatrix with either 2 rows or 2 columns. Rational realization conjecture. There exi…

Let $A_i$, $B_j$, $C_k$, $D$, and $F$ be generic $3\times3$ matrices, and let $X\in\mathcal{M}_3(\overline{K})$. Define … The maximal quadratic equation conjecture. The equation…

Let $K$ be a field, let $g$ be a nonzero polynomial in the noncommuting indeterminates $u_1,\ldots,u_{k+1}$ such that … for every $i\leq k$, the indeterminate $u_i$ appears exactly…

Let $K$ be the base field, let $n=3$, and let $B,C,D$ be generic $3\times 3$ matrices. Consider Eq., the nonsymmetric generic matrix equation in the unknown matrix…

Let equation be the complex matrix equation and equation its associated real matrix equation. Denote the real representation of a solution $X$ by $X_{\sigma}$. Uniqueness-correspon…

Let equation be the complex matrix equation considered in the paper, and let equation be its associated real matrix equation. If $\boldsymbol{Y}$ is the general real solution of eq…

Let $p$ and $q$ be the matrices defined by … and let $Q$ be determined perturbatively by … Here ${\cal S}_{p,q}$ denotes the operation that reorders each term of a power-series exp…

Let $\mathcal{L}_0$ and $\mathcal{L}_1$ be matrices, and consider the quadratic matrix equation $X^2-\mathcal{L}_1X-\mathcal{L}_0=0$. A universal, possibly multivalued matrix funct…