4 problems

Let $A_i\in\mathcal F^{m_{i+1}\times m_i}$ for $i=1,2,\ldots,k$, and let $A_i^-$ denote an inner inverse satisfying $A_iA_i^-A_i=A_i$. Conjecture 30.14. The minimum rank of the pro…

Let $A\in\mathcal F^{m\times m}$ be a square matrix, and let $A^-$ be a generalized inverse of $A$. Conjecture 21.20. For every choice of $A^-$, the rank of the commutator … is eve…

Let $A\in{\cal F}^{m\times n}$ and let $\lambda_1,\ldots,\lambda_k\in\mathcal F$ satisfy $\lambda_i\neq\lambda_j$ for distinct indices. A generalized inverse $(\lambda_i I_m-A)^-$…

Let $A=A_s+epsilon A_i\in\mathbb{DR}^{m\times n}$ be a dual matrix with $\operatorname{rank}(A_s)=k$. A rank-$k$ decomposition of $A$ means a factorization $A=BC$ with…