Parallelogram-law-type identities
type: Item
instance of: scholarly article
title: Parallelogram-law-type identities
zbMATH Open document ID: 0831.51017
DOI: 10.1016/0024-3795(93)00301-F
author: Ömer Eğecioğlu
published in: Linear Algebra and its Applications
publication date: 1995-10-09T00:00:00Z
review text: The parallelogram law in the complex plane is \(2(|a|^2+ |b|^2)= |a+ b|^2+ |a- b|^2\); it relates the lengths of the diagonals of a parallelogram with vertices \(0\), \(a\), \(b\) and \(a+ b\) to the lengths of its sides. In three theorems, the author proves several generalizations of this formula; they all refer to arbitrary complex numbers \(z_1,\dots, z_n\) (instead of \(a\), \(b\)) together with a fixed set \(A= \{\alpha_1,\dots, \alpha_n\}\) of complex numbers with \(\alpha_1+\cdots+ \alpha_n= 0\). Special choices of \(n\) and \(A\) lead to formulas involving sums of roots of unity, trigonometric functions, binomial coefficients, and permutations over the symmetric and the alternating groups (of degree \(n\)).
Mathematics Subject Classification ID: 51N20
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 803707
zbMATH Keywords: parallelogram law
zbMATH Keywords: permutation
zbMATH Keywords: complex plane
zbMATH Keywords: symmetric group
zbMATH Keywords: alternating group
MaRDI profile type: Publication
cites work: Q5782157
cites work: On representations of the braid group
cites work: Q5614003
cites work: Q5577154
cites work: Q4275389
cites work: Q5767179
description: scientific article; zbMATH DE number 803707