79 problems

For $m,n\in\mathbb{N}$, let $A_n^{(m)}$ be the $n\times n$ matrix defined by … Here $\delta_{ij}$ is the Kronecker delta. Sun's determinant-form conjecture. Sun conjectured that ……

Let a symmetric palindromic Toeplitz matrix have a limiting eigenvalue-gap distribution, and let a real symmetric random-matrix ensemble have independent matrix elements drawn from…

Let $l,g$ be two even functions with $g>0$ on $(0,\pi)$, and suppose that $f\equiv l/g$ is monotone increasing over $(0,\pi)$. Set $X_n\equiv T_n^{-1}(g)T_n(l)$ for all $n$. Let th…

Let $R\geq 2$ and let … where $t_1,\ldots,t_R$ are distinct points on ${\bf T}$, $0<\operatorname{Re}\alpha_r<1/2$, $-1/2<\operatorname{Re}\beta_r\leq 1/2$, and…

Let $A_N$ be an $N\times N$ real symmetric Toeplitz matrix from the Toeplitz ensemble, and consider its eigenvalues after normalization so that the local spacings between adjacent…

Let $M_N(\phi)$ denote the symmetric Toeplitz + Hankel matrix associated with the generating function $\phi$, and let $t_{\beta,\theta}$ denote the singularity factor used in the p…

Let $n\geq 2$, and let $\mathcal{T}_n$ be the affine variety of $n\times n$ Toeplitz matrices. Write $\mathbf{FL}_{\mathrm{max}}(\mathcal{T}_n)$ for the maximal $\mathcal{T}_n$-fac…

Randomised twisted Toeplitz spectral-measure conjecture. The measures $\mu_n$ converge weakly to

Banded perturbation conjecture. The measures $\mu_n$ converge weakly to

Let $k$ be a positive integer and let … where, for every $i,j=1,\dots,k$, $f_{i,j}\in L^{\infty}([-i,i]])$ is real-valued almost everywhere, and either $T_n(f_{i,j})$ is Hermitian…

Johnson's conjecture. For every $n\ge 2$ and every Toeplitz matrix $A\in M_n(\mathbb{R})$ satisfying $A+A^{\top}=2J_n$,

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 $n\in\mathbb{N}$,

Let $T_n$ be the Cauchy-Toeplitz matrix … For $1\leq p,q\leq\infty$, let $\|T_n\|_{p,q}$ denote its $\ell_{p,q}$ norm. Bozkurt's conjecture. The inequalities … and … are valid. The…

Let $B(z)$ be a real $k\times k$ matrix-valued symbol. Define … where $f(z,\lambda)$ is the characteristic function of $B$, and let $\Lambda_0(B)$ denote the set of limiting spectr…

For each integer $K\geq1$, let $F_K(t)$ be the $K\times K$ Hermitian matrix-valued Fourier series associated with the block Toeplitz matrices $A(N,K)$, … Let…

Let $X$ be a general Toeplitz–Robinson matrix in the seriation problem, where the goal is to recover its latent ordering from noisy observations. Computational-statistical gap conj…

Let the sample matrices be … where $\mathbf{\xi}_k$ are independent copies of a random vector $\mathbf{\xi}$ whose entries are i.i.d. copies of a subgaussian random variable $\xi$…

Let $r$ be a continuously differentiable real function depending on a parameter $t$ in a real open interval containing the origin. Fix $k\in\mathbb{Z}$ and define the moments and t…

Let $X$ and $Y$ be arbitrary real Toeplitz matrices, let $\langle X,Y\rangle$ denote their Frobenius inner product, and let $\|\cdot\|$ denote the Frobenius norm. Lu–Wenzel conject…

Let $k\geq 2$ and consider a tridiagonal $k$-Toeplitz matrix with symbol whose eigenvalues are denoted by $\lambda_j(z)$. The eigenvectors are said to exhibit exponential decay whe…

Let $n$ be the matrix size and let $m\leq n$ be the model parameter, with fixed parameters $\delta$ and $a$. Spectral scaling conjecture. For both model 1 and model 2, when $n$ and…

Let $n$ be the matrix size, let $m\geq 1$ be fixed, and let $S_{\delta J}^{(2)}(m)$ denote model 2. Let $\lambda_0$ be the distinguished point included in its pseudospectrum, and l…