{"id":"hexagon/2610.00671","pdf":"https://hexagonmath.org/pdf/2610.00671v1","url":"https://hexagonmath.org/abs/2610.00671v1","tags":["msc:68W30","msc:68W40","msc:15A69"],"score":0,"title":"Improved basis-change transformations for $2 \\times 2$ matrix multiplication","domain":"cs.DS","source":"hexagon","authors":["Hugo Nielsen"],"reviews":0,"subject":"cs.DS","summary":"To the best of our knowledge, we present a state-of-the-art change-of-basis algorithm for $2 \\times 2$ matrix multiplication under the Karstadt--Schwartz-style basis change regime [1]. We do this by introducing basis-change matrices $B_U$, $B_V$, and $B_W$ that each require only a single addition and hence give rise to a matrix-multiplication algorithm performing $5n^{\\log_2{7}} - 4n^2 + \\frac{3}{4}n^2 \\log_2{n},$ arithmetic operations when multiplying two $n \\times n$ matrices, with $n$ a power of $2$. This improves on the previous best bound, which had last term $\\frac{3}{2}n^2\\log_2{n}$.","category":"research","comments":0,"disputed":false,"division":"cs","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-10-31","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}