9 problems

Let $\widetilde{A}$ be an $n\times n$ matrix with $\|\widetilde{A}\|_2\leq 1$, and let $A$ be a Gaussian perturbation of $\widetilde{A}$ with variance $\sigma^2\leq1$. Let $U$ be t…

Let $A=B+iC\in\mathcal A_n$, where $B$ and $C$ are the real and imaginary parts of an accretive-dissipative matrix, and suppose that … Here $\kappa$ denotes the condition number an…

Let $A=B+iC$ be a complex symmetric matrix, where $B,C\in\mathbb R^{n\times n}$ are real symmetric positive definite. Such a matrix is called a Higham matrix. Let $\rho_n(A)$ denot…

Maximum growth-factor conjecture. The largest possible growth factor for a $5\times 5$ real matrix under complete pivoting is exactly the lower bound proved in this paper, namely…

Let $g_n(\mathbb{R})$ denote the maximum growth factor under complete pivoting over all nonsingular $n\times n$ real matrices. Linear growth-factor conjecture. One has … Wilkinson'…

Optimality conjecture. This lower bound is optimal on $\operatorname{O}(n)$, namely

Superlinear growth conjecture. The maximum growth factor is superlinear:

For each positive integer $n$, let $g\big[\mathbf{CP}_n(\mathbb{R})\big]$ be the maximum growth factor for complete pivoting over real $n\times n$ matrices. Complete-pivoting thres…

Let $g\big[\mathbf{CP}_n(\mathbb{R})\big]$ denote the maximum growth factor for complete pivoting over real $n\times n$ matrices. Wilkinson's conjecture. … with equality achieved o…