An Exact Bound for the Inner Product of Vectors in Cn

An exact upper bound of 12 is shown for the difference between the inner product of vectors in Cn. This bound is attained when the vectors are unit vectors. The inequality provided in the proposition can be seen as a lower bound on the modulus of the inner product. It is reminiscent of the reverse C...

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Bibliographic Details
Published in:The American mathematical monthly 2024-08, Vol.131 (7), p.627
Main Author: Pinelis, Iosif
Format: Article
Language:English
Subjects:
Cauchy problems
Euclidean space
Income inequality
Inequality
Lagrange multiplier
Linear equations
Lower bounds
Upper bounds
Online Access:Get full text
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Summary:An exact upper bound of 12 is shown for the difference between the inner product of vectors in Cn. This bound is attained when the vectors are unit vectors. The inequality provided in the proposition can be seen as a lower bound on the modulus of the inner product. It is reminiscent of the reverse Cauchy-Schwarz inequality. The proof of the proposition involves analyzing Lagrange multipliers and finding a clever solution. The case where the vectors are in Rn is considered first, and then the general case with unit vectors in Cn is addressed. The problem is reduced to proving a claim, which is easily proven using the Cauchy-Schwarz inequality. No potential conflict of interest is reported by the author.
ISSN:0002-9890
1930-0972
DOI:10.1080/00029890.2024.2344412