14 problems

Improved eta-expanded theorem. The thesis of Theorem …

Ceiling-bound conjecture. For $n>1$,

Quasi-Toeplitz row-equivalence conjecture. Every $n\times n$ matrix is row equivalent to a quasi-Toeplitz matrix.

Bounded invertible reduction conjecture. There exists a constant $C>0$ such that every matrix can be expressed as a product of an invertible matrix and $C$ Toeplitz matrices. In pa…

Ye–Lim's conjecture. For all $n\in\mathbb{N}$,

Consider the $\ell_1$-norm rank-one symmetric matrix factorization problem and initialize the subgradient method randomly, with an initialization distribution absolutely continuous…

Let $n$ be an even natural number greater than $2$. A quasi-prime matrix is a matrix that is the product of two quasi-invertible semi-prime matrices, where a semi-prime matrix is a…

Let $\mathbf{H} \in \mathbb{R}^{n\times n}$ be a positive definite Hankel matrix with bit complexity $\ell$ and condition number bounded by $2^\ell$. An inverse symmetric positive…

Let $\mathbf{H} \in \mathbb{R}^{n\times n}$ be a positive definite Hankel matrix with bit complexity $\ell$. A symmetric positive factorization algorithm should find a representati…

Cossu–Zanardo necessity conjecture. If this matrix can be written as a product of two idempotent matrices, then it must satisfy the Cossu–Zanardo conjecture.

Cossu–Zanardo conjecture. There exist $a,b,c\in\mathbb{Z}[\sqrt{D}]$ with $a(1-a)=bc$ such that

Let $A$ be a nonnegative matrix, let $A\otimes A$ denote its Kronecker square, and write $\operatorname{rank}(A)$ for its ordinary rank. Self-Kronecker-product conjecture. … The so…

Let $P$ be a finite permutation matrix of bandwidth $w>0$. A greedy bubble matrix of $P$ is a product of reducing and overtaking swaps of $P$ such that $P$ and the resulting matrix…

Let $P$ be a finite permutation matrix of bandwidth $w>0$, where bandwidth $1$ means tridiagonal. A bandwidth--1 permutation matrix has nonzero entries only on the main diagonal an…