13 problems

Let $V$ be a finite-dimensional vector space over $\mathbb K$ of positive dimension, and let $A,A^*,A^\varepsilon:V\to V$ be linear transformations. A Leonard pair is a pair of tra…

Let $P$ be an $n\times n$ tridiagonal stochastic matrix. Its eigenvalues are real when $P$ is reversible, and the second largest eigenvalue means the largest algebraic eigenvalue a…

Let $A$ be an irreducible $4\times4$ tridiagonal stochastic matrix of the form … where $0\leq\alpha,\beta,\gamma,\delta,\phi,\kappa\leq1$, $\beta+\gamma\leq1$, and…

Xin–Zhang's determinant conjecture. The determinant of $C$ is

Let $P_n$ be the characteristic polynomial of the tridiagonal random matrix model with $V_1\equiv a$, where $|W_j|$ has a non-degenerate distribution supported on $\mathbb Z_{>0}$…

Let $N$ be a positive integer and let $T^-$ and $T^+$ be the $(N-1)\times(N-1)$ tridiagonal matrices with … for $j=1,\ldots,N-1$, and $T^\pm_{j,j+1}=T^\pm_{j+1,j}=1$. Assume $a>0$.…

Let $A\in\mathbb{R}^{n\times n}$ be a nonsingular matrix whose inverse has only nonzero elements in the lower triangle. Let $X$ be the inverse computed numerically by algorithm (KW…

Let $A\in\mathbb{R}^{n\times n}$ be a tridiagonal matrix, and let $X$ be its inverse computed using the proposed algorithm. Let $\varepsilon$ denote the machine precision, let…

Chandler-Wilde–Chonchaiya–Lindner's density conjecture. The set $\sigma_\infty$ is dense in $\Sigma$. The conjecture concerns whether the eigenvalues of all finite tridiagonal sign…

Let a finite tridiagonal sign matrix have entries $+1$ or $-1$ on its first sub- and superdiagonals and zero entries elsewhere. Let a periodic tridiagonal sign operator act on the…

Let $n$ be a nonnegative integer, and consider the $(n+1)\times(n+1)$ Sylvester–Kac matrix with zero main diagonal, superdiagonal $(1,2,\ldots,n)$, and subdiagonal $(n,\ldots,2,1)$…

Let $\mathcal{T}_n$ be the set of $n\times n$ tridiagonal real matrices whose sub-diagonal entries are negative, super-diagonal entries are positive, the $(1,1)$ entry is negative,…

Meixner–Schäfke conjecture. The convergence radii satisfy