13 problems

Variable-size eigenvector decomposition conjecture. For any $3\times3$ octonionic Hermitian matrix, the associativity relation should hold when suitably rewritten for a set consist…

Let $A=A_s+epsilon A_i\in\mathbb{DR}^{m\times n}$ be a dual matrix with $\operatorname{rank}(A_s)=k$. A rank-$k$ decomposition of $A$ means a factorization $A=BC$ with…

Let $\mathcal{A}$ be a finite-dimensional $\ast$-algebra over $\mathbb{R}$, and let $H\in\mathcal{A}$ be self-adjoint, meaning $H^{\ast}=H$. Let $k\in\mathbb{N}$ be maximal among t…

Conjecture. All companion $n\times n$ matrices over $\mathbb{F}$ are the sum of a potent matrix and a square-zero matrix over $\mathbb{F}$ whose summands commute if and only if…

Let $(\mathcal A,*)$ be an associative, unital, finite-dimensional $*$-algebra over $\mathbb R$. The spectral monoid $\overline{\operatorname{Her}}(\mathcal A,*)$ is the monoid of…

Let $Z\in\mathbb{Q}^{n\times n}$ be a nonzero quaternion skew-symmetric matrix. For $n>2$, call $Z$ basic if there is no unitary matrix $U$ such that $UZU^*=\Sigma$, where $\Sigma$…

DODD existence conjecture. The $d\times d$ 0-inflation of any matrix $\mathbf{X}\in\mathbb{R}^{m\times n}$ admits a DODD for some sufficiently large $d\geq m,n$. Thus every such ma…

Applications conjecture. This simultaneous decomposition will also play an important role in signal processing and linear modelling.

Let $U$ be a unitary matrix. A unitary diagonal matrix is a diagonal matrix whose diagonal entries have modulus $1$, and a unitary doubly stochastic matrix is a unitary matrix whos…

Let $U\in U(n)$ be generic, and consider its normal forms $U=e^{i\varphi}D_1AD_2$, where $D_1$ and $D_2$ are unitary diagonal matrices with $(D_i)_{11}=1$ and $A$ has all row and c…

Let $U$ be an $n\times n$ unitary matrix. A matrix is doubly stochastic when all entries in each row and column sum to one, and a matrix is unitary diagonal when it is both diagona…

Convergence conjecture. The numerical algorithm converges to a unit line-sum matrix, that is, to a member of $\operatorname{XU}(n)$.

Two-factor decomposition conjecture. Even the stronger bound $p\leq 2$ should hold.