29 problems

Let $\mathbb{A}=(A,B)\in\mathbf{G}^2$, where $\mathbf{G}=\{X\in\mathrm{GL}(2,\mathbb{R}):\det X>0\}$, and let $\hat{\rho}(\mathbb{A})$ be the joint spectral radius, defined by … A…

Binary-matrix finiteness conjecture. Every pair of binary matrices has the finiteness property.

Let a set of matrices have the finiteness property when its joint spectral radius is attained by some finite product. Finiteness conjecture. It was conjectured that all sets of mat…

Let $A_0$ and $A_1$ be the matrices defining the joint spectral radius $\rho$, and let $\Pi_n=A_{i_1}\cdots A_{i_n}$ be a product of length $n$. Finiteness conjecture. Numerical ob…

Local Hölder continuity conjecture. In general, the joint spectral radius is locally Hölder continuous on $\mathcal{H}(d)$, with order

Locally uniform trajectory-bounds conjecture. There is a constant $\Theta_d>0$, depending only on $d$ and the norm, such that for every $\mathcal{M}\in\mathcal{H}(d)$ with…

Maesumi's conjecture. Lebesgue-almost every finite family of matrices has an SMP. The conjecture was proposed after the finiteness conjecture was refuted by Bousch and Mairesse; th…

Undecidability conjecture. It is undecidable to check whether

Undecidability conjecture. It is undecidable to check whether $\rho\geq 1$ for the joint spectral radius $\rho$.

Let $\Sigma=\{A_0,A_1,A_2,A_3\}$ be the four matrices associated with the square billiard table, and let $\rho(\Sigma)$ denote the joint spectral radius of $\Sigma$. A maximizing m…

Algorithmic unsolvability conjecture. The recognition problem for CP families with rational matrix entries is algorithmically unsolvable.

A binary matrix is a matrix whose entries belong to $\{0,1\}$. For a finite matrix family, the finiteness property means that some finite product attains the joint spectral radius.…

Rational-matrix finiteness conjecture. Every finite family of rational matrices has the finiteness property. The source says this has only been conjectured and gives no resolution.

Let $\mathcal F$ be any finite set of square matrices with rational entries, and let $\overline\rho_k(\mathcal F)$ denote the maximum normalized spectral radius among products of l…

Let the ambient space consist of finite families of complex square matrices, equipped with its natural measure, and let the finiteness property mean that some finite product attain…

Blondel et al.'s conjecture. The set of parameters $\alpha\in[0,1]$ for which $\mathcal F_\alpha$ does not satisfy the finiteness property has measure zero. The conjecture is prese…

Let $\mathcal B$ be the space of bounded families of matrices, $\mathcal R$ the subset of reducible families, and $\mathcal D\subsetneq\mathcal R$ the subset of defective families.…

Let ${\mathcal{A}}$ be a finite family of real matrices. The invariant polytope algorithm computes the joint spectral radius $\rho({\mathcal{A}})$ in finite time for a full-Lebesgu…

Maximal-growth trajectory conjecture. For a generic discrete-time system, there exists a trajectory $x(k)$ such that

Resonance-degree conjecture. For a generic discrete-time system, there exist constants $C_1,C_2>0$ such that

Let $\mathbb{A}\in\{0,1\}^{K\times K}$ be any matrix, and let … with its natural subspace topology. Here spectral finiteness restricted to $\mathbb{A}$ means that the joint spectra…

Let $T\colon W\rightarrow W$ have the closing by periodic orbits property, and let $\boldsymbol{C}\in\operatorname{C}(W,\mathbb{R}^{d\times d})$ be a continuous matrix-valued funct…

Let $\mathcal{F}$ be a pair of $n\times n$ sign-matrices, that is, matrices whose entries are signs. The restricted finiteness conjecture. The finiteness property should hold for e…

Let $\boldsymbol{S}=\{S_1,S_2\}\subset\mathbb{R}^{d\times d}$, with $d\geq 4$, be an arbitrary pair satisfying condition (1.3a), namely that the matrices share a common non-strict…

Let $r$ be an odd integer, and let $A_1,A_2$ be the $(r-1)\times(r-1)$ matrices defined by … and $(A_s)_{ij}=0$ otherwise, for $s=1,2$. A product is an s.m.p. if it is a spectrum-m…