On Hastings' approach to Lin's Theorem for Almost Commuting Matrices

Lin’s theorem states that for all $ϵ>0$, there is a $\delta >0$ such that for all $n\ge 1$ if self-adjoint contractions $A,B \in M_n($\mathbb{C}$)$satisfy$|[A,B]|< \delta$ then there are self-adjoint contractions $A^{\mathrm{\prime }},B^{\mathrm{\prime }}\in M_n(\mathbb{C})$ with $[A^{\mathrm{\prime }},B^{\mathrm{\prime }}]=0$ and $\mathrm{\parallel }A-A^{\mathrm{\prime }}\mathrm{\parallel },\mathrm{\parallel }B-B^{\mathrm{\prime }}\mathrm{\parallel }<ϵ$. We present fully explained and corrected details of the approach in arXiv:0808.2474, which was the first version of Lin’s theorem to provide asymptotic estimates. We also apply this method to the case where $B$ is a normal matrix with spectrum lying in some nice 1-dimensional subset of $\mathbb{C}$.