Perturbations of an Invertible Matrix: the Invertible Matrices Form an Open Set and Inversion Is Continuous

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $n\in\mathbb{N}$, with the square real matrices $\mathcal{M}_{n}(\mathbb{R})$, invertible matrices $C$ and their inverses $C^{-1}$, the operator norm $\lVert C\rVert$ of a matrix $C$, and the distance $d_{n}$, a metric on $\mathcal{M}_{n}(\mathbb{R})$ by The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence §metric. Let $\mathrm{GL}_{n}(\mathbb{R})=\{C\in\mathcal{M}_{n}(\mathbb{R}):C\text{ is invertible}\}$ be the set of the invertible matrices in $\mathcal{M}_{n}(\mathbb{R})$ of The Invertible Matrices and the Positive Semidefinite Matrices Form Sets §invertible.

¶ Let $A\in\mathcal{M}_{n}(\mathbb{R})$ be invertible. If $B\in\mathcal{M}_{n}(\mathbb{R})$ and $\lVert A^{-1}\rVert\,d_{n}(A,B)<1$, then $B$ is invertible and

$$d_{n}(A^{-1},B^{-1})\le\frac{\lVert A^{-1}\rVert^{2}\,d_{n}(A,B)}{1-\lVert A^{-1}\rVert\,d_{n}(A,B)} .$$

¶ $\mathrm{GL}_{n}(\mathbb{R})$ is open in $(\mathcal{M}_{n}(\mathbb{R}),d_{n})$.

¶ Let $A\in\mathcal{M}_{n}(\mathbb{R})$ be invertible. For every real $\varepsilon>0$ there is a real $\delta>0$ such that every $B\in\mathcal{M}_{n}(\mathbb{R})$ with $d_{n}(A,B)<\delta$ is invertible and satisfies $d_{n}(A^{-1},B^{-1})<\varepsilon$.

¶ Let $d_{\mathrm{GL}}$ be the restriction of $d_{n}$ to $\mathrm{GL}_{n}(\mathbb{R})\times\mathrm{GL}_{n}(\mathbb{R})$, a metric on $\mathrm{GL}_{n}(\mathbb{R})$ by Subsets of a Metric Space: the Restricted Metric, Sequences, Limits in Closed Subsets and Completeness of Closed Subsets §restriction. There is exactly one map $f:\mathrm{GL}_{n}(\mathbb{R})\to\mathcal{M}_{n}(\mathbb{R})$ with $f(B)=B^{-1}$ for every $B\in\mathrm{GL}_{n}(\mathbb{R})$, the map $B\mapsto B^{-1}$; and $f$, from $(\mathrm{GL}_{n}(\mathbb{R}),d_{\mathrm{GL}})$ to $(\mathcal{M}_{n}(\mathbb{R}),d_{n})$, is continuous.