# 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}
}
• Jessica E. Banks, +1 author N. Srivastava
• Published 2020
• Mathematics, Computer Science
• 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
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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