User:Fawly/Computational complexity of matrix multiplication: Difference between revisions
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There is a trivial lower bound of {{tmath|\omega \ge 2}}. Since any algorithm for multiplying two {{math|''n'' × ''n''}}-matrices has to process all {{math|2''n''<sup>2</sup>}} entries, there is a trivial asymptotic lower bound of {{math|Ω(''n''<sup>2</sup>)}} operations for any matrix multiplication algorithm. Thus {{tmath|2\le \omega < 2.373}}. It is unknown whether {{tmath|\omega > 2}}. The best known lower bound for matrix-multiplication complexity is {{math|Ω(''n''<sup>2</sup> log(''n''))}}, for bounded coefficient [[Arithmetic circuit complexity|arithmetic circuits]] over the real or complex numbers, and is due to [[Ran Raz]].<ref>{{cite journal | last1 = Raz | first1 = Ran | author-link = Ran Raz | year = 2002| title = On the complexity of matrix product | journal = Proceedings of the Thirty-fourth Annual ACM Symposium on Theory of Computing | pages = 144 | doi = 10.1145/509907.509932 | isbn = 1581134959 | s2cid = 9582328 }}</ref>
=== Rectangular matrix multiplication ===▼
<ref>{{Citation|last=Gall|first=Francois Le|title=Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor|date=2018-01-01|url=https://epubs.siam.org/doi/10.1137/1.9781611975031.67|work=Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)|pages=1029–1046|series=Proceedings|publisher=Society for Industrial and Applied Mathematics|doi=10.1137/1.9781611975031.67|access-date=2021-05-23|last2=Urrutia|first2=Florent|arxiv=1708.05622}}</ref>▼
==Related complexities==
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Problems that have the same asymptotic complexity as matrix multiplication include [[determinant]], [[matrix inversion]], [[Gaussian elimination]] (see next section). Problems with complexity that is expressible in terms of <math>\omega</math> include characteristic polynomial, eigenvalues (but not eigenvectors), [[Hermite normal form]], and [[Smith normal form]].{{citation needed|date=March 2018}}
▲=== Rectangular matrix multiplication ===
▲<ref>{{Citation|last=Gall|first=Francois Le|title=Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor|date=2018-01-01|url=https://epubs.siam.org/doi/10.1137/1.9781611975031.67|work=Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)|pages=1029–1046|series=Proceedings|publisher=Society for Industrial and Applied Mathematics|doi=10.1137/1.9781611975031.67|access-date=2021-05-23|last2=Urrutia|first2=Florent|arxiv=1708.05622}}</ref>
===Matrix inversion, determinant and Gaussian elimination===
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