Pseudospectral Shattering, the Sign Function, and Diagonalization in Nearly Matrix Multiplication Time

@article{Banks2020PseudospectralST,
  title={Pseudospectral Shattering, the Sign Function, and Diagonalization in Nearly Matrix Multiplication Time},
  author={Jessica E. Banks and Jorge Garza-Vargas and Archit Kulkarni and N. Srivastava},
  journal={2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)},
  year={2020},
  pages={529-540}
}
We exhibit a randomized algorithm which given a square matrix $A\in \mathbb{C}^{n\times n}$ with $\Vert A\Vert\leq 1$ and $\delta > 0$, computes with high probability an invertible $V$ and diagonal $D$ such that \begin{equation*}\Vert A-VDV^{-1}\Vert\leq\delta\end{equation*} in $O(T_{\text{MM}}(n)\log^{2}(n/\delta))$ arithmetic operations on a floating point machine with $O(\log^{4}(n/\delta)\log n)$ bits of precision. The computed similarity $V$ additionally satisfies $\Vert V\Vert\Vert V^{-1… Expand

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