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For infinite matrices, Jaffard's theorem states that if $(A(k,l))_{k,l\in \mathbb{Z}}$ is invertible and satisfies $$ A(k,l) \leq C (1+\left|k-l\right|)^{-r}, $$ for some $C>0$, then $$ A^{-1}(k,...

Let $k\in\mathbb{N}\cup\{0\}$, let $\Omega\subseteq\mathbb{R}^n$ be open, connected and let $G\in C^k(\Omega;\mathbb{R}^{n\times n})$ satisfy $$ \operatorname*{rank}G(x)= \operatorname*{rank}G(y),\...

Let $$M = \begin{bmatrix} A & B \\ B & C \end{bmatrix}$$ in which $A$, $B$ and $C$ are $3 \times 3$ matrices being also symmetric. In fact, they are quite similar, just differing on a single ...