35 problems

Let $A$ be a nonnegative substochastic matrix, meaning … for all $i$ and $j$. Suppose that above the diagonal, $A$ has nonzero entries only at distance $1$ from the diagonal, while…

Rank-deficient permutation coverage conjecture. For $V$ generic Toeplitz, every permutation $P$ for which

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…

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

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

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…

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

Let $S_{\delta J}^{(2)}(m)$ denote the model 2 matrix, with matrix size $n$, parameters $\delta$ and $a$, and let $\lambda_0$ be the distinguished eigenvalue of the original system…

The eigenvalue decomposition conjecture. The positive eigenvalues satisfy

Let $A$ be a nonnegative substochastic matrix, meaning … for all $i$ and $j$. Suppose that above the diagonal, $A$ has nonzero entries only at distance $1$ from the diagonal, while…

Consider the Markov chain on $\mathbb{N}$ in which every vertex $i$ has an edge of weight $r$ to $i-1$ and edges of weight $c_l$ to $i+l$ for $0\leq l\leq k$, with … Let $p_l$ be t…

Extreme-eigenvalue expansion conjecture. There exists an integer $k\geqslant1$ such that

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…

Let $A$ satisfy the conditions of Theorem 3.1, and let $A'$ be the matrix constructed in Lemma 3.4. For the Toeplitz coefficients $a_j$, define the limiting measure $\mu$ as the di…

Let $L_{n,N}^{(1)}$ and $L_{n,N}^{(2)}$ denote the two types of closed eigenvalue contour lines associated with the Kac–Murdock–Szegő matrix of size $n$, and let…

Let $g_m(x)=\left(2\sin\frac{x}{2}\right)^{2m}$ for an integer $m\geq 3$, and let $\lambda_{n,j}$ be the eigenvalues of the Toeplitz matrices generated by $g_m$. Let $d_k$ denote t…

Let $T$ be the Toeplitz matrix, let $A_{jj}$ denote the indicated diagonal rank-one perturbation, and let $V$ denote the perturbation appearing in the source. Consider the deformat…