Abstract
This note mainly aims to improve the inequality, proposed by Böttcher and Wenzel, giving the upper bound of the Frobenius norm of the commutator of two particular matrices in ℝn×n. We first propose a new upper bound on basis of the Böttcher and Wenzel’s inequality. Motivated by the method used, the inequality ‖XY − XY‖ 2 F ≤ 2‖X‖ 2 F ‖Y‖ 2 F is finally improved into
. In addition, a further improvement is made.
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A. Böttcher and D. Wenzel, “How big can the commutator of two matrices be and how big is it typically?,” Linear Algebra Appl. 403, 216–228 (2005).
L. László, “Proof of Böttcher and Wenzel’s conjecture on commutator norms for 3-by-3 matrices,” Linear Algebra Appl. 422(2–3), 659–663 (2007).
S.-W. Vong and X.-Q. Jin, “Proof of Böttcher and Wenzel’s conjecture,” Oper. Matrices 2(3), 435–442 (2008).
A. Böttcher and D. Wenzel, “The Frobenius norm and the commutator,” Linear Algebra Appl. 429(8–9), 1864–1885 (2008).
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Published in Russian in Matematicheskie Zametki, 2010, Vol. 87, No. 6, pp. 935–940.
The text was submitted by the authors in English.
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Wu, YD., Liu, XQ. A short note on the Frobenius norm of the commutator. Math Notes 87, 903–907 (2010). https://doi.org/10.1134/S0001434610050305
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DOI: https://doi.org/10.1134/S0001434610050305