28 problems

Optimal low-rank covariance approximation conjecture. For almost any positive definite matrix $A$, for every iteration $m\leq n$, there exist $d(m)\ll n$ and $r(d)\leq d(m)$ such t…

Let $\Lambda_{1},\dotsc,\Lambda_{d}\in\mathbb{R}^{b\times b}$ be real diagonal matrices, and assume that $\Lambda_i$ and $\Lambda_j$ have disjoint spectra for $1\leq i<j\leq d$. Le…

Rayleigh–Ritz mixed-type majorization error bounds conjecture. The following weak majorization inequalities hold:

Let ${\mathcal X}$ and ${\mathcal Y}$ be subspaces of a Hilbert space ${\mathcal H}$ with ${{\rm \dim}} \mathcal X={{\rm \dim}} \mathcal Y$, and let $A$ be Hermitian. For a vector…

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 $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^7}}$ denote the closure of the polynomial set obtainable with sev…

Let $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^6}}$ denote the closure of the polynomial set obtainable with six…

Let $m>2$ and let $\Pi^*_{2^m}$ denote the polynomial set associated with evaluation schemes using $2^m$ matrix-matrix multiplications. Numerical computations suggest that generic…

Collatz convergence-rate conjecture. What is the condition such that both sequences $\{\underline{\lambda}_k\}$ and $\{\overline{\lambda}_k\}$ converge to the same dual number…

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…

The eigenvalue decomposition conjecture. The positive eigenvalues satisfy

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…

Let $A_0=U_0T_0U_0^*\in\mathbb{C}^{n\times n}$, where $U_0$ is unitary and $T_0$ is upper triangular. A factorization of $A$ as $A=UTU^*$ is a Schur decomposition when $U$ is unita…

DODD existence conjecture. The $d\times d$ 0-inflation of any matrix $\mathbf{X}\in\mathbb{R}^{m\times n}$ admits a DODD for some sufficiently large $d\geq m,n$. Thus every such ma…

Let $A$ be the matrix, let $B$ be the output of incomplete selected inversion, and let $F$ denote the error matrix in the notation of the paper's selected-inversion setup. Write…

Let $H$ be the matrix and let $E$ denote the error matrix in the notation of the paper's selected-inversion setup. Write $\mathrm{level}(i,j)$ for the level of fill between indices…

Let $A\in\mathbb{R}^{m \times m}$, let $\\|\cdot\\|$ be a norm on $\mathbb{R}^m$, and let $P,Q,R$ be the factors output by the arbitrary-norm rank-revealing QR algorithm described…

Finite-matrix convergence conjecture. There is a (modest) constant $C\in\mathbb R$ and a technique of associating a measure $\sigma$ with compact support and continuous potential t…

Let \text{text{$\mathversion{bold}$ A }}$\in\mathbb{IR}^{n\times n}$ be an interval matrix. A submatrix is real when all of its entries are degenerate intervals, and \text{te…

Let $A$ be the symmetric positive-definite matrix defining the non-standard inner product, let $Z$ be the input matrix with $m$ rows, let $n$ be the number of columns, and let…

Let $A$ be nonsingular. Define the non-orthogonal coarse-grid correction using transfer operators … The correction is stable in the sense that … for some constant $C$ independent o…

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…