Quadratic formulas for quaternions
type: Item
instance of: scholarly article
title: Quadratic formulas for quaternions
zbMATH Open document ID: 1011.15010
DOI: 10.1016/S0893-9659(02)80003-9
published in: Applied Mathematics Letters
publication date: 2002-08-15T00:00:00Z
review text: Let \(\mathbb{R}\) denote the field of real numbers and \(\mathbb{H}\) denote the algebra of real quaternions. The authors of the paper provide the explicit formulas for computing all roots of a quadratic polynomial of the form \(x^2+bx+c\), for all \(b,c\in \mathbb{H}\). The general situation is divided into several particular cases depending on the relations between parameters \(b\) and \(c\). In each case the direct expressions of all roots through the parameters \(b\) and \(c\) are derived. As a corollary, the necessary and sufficient conditions on parameters \(b,c\) that guarantee the existence of exactly one, two, or infinitely many different roots are given. The results of this paper unify several particular results on this subject [see, \textit{S. Zhang} and \textit{D. Mu}, J. Math. Res. Expo. 14, No. 2, 260-264 (1994; Zbl 0815.16007), \textit{R. M. Porter}, J. Nat. Geom. 11, No. 2, 101-106 (1997; Zbl 0869.11024), \textit{I. Niven}, Am. Math. Mon. 48, 654-661 (1941; Zbl 0060.08002)].
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 11D09
Mathematics Subject Classification ID: 12D10
zbMATH DE Number: 1781915
zbMATH Keywords: quatornions
zbMATH Keywords: quadratic equations
author: Liping Huang
author: Wasin So
reviewed by: Alexander Guterman
MaRDI profile type: Publication
cites work: Quaternions and matrices of quaternions
cites work: Q4311691
cites work: Q3124430
cites work: Equations in Quaternions
cites work: On left eigenvalues of a quaternionic matrix
cites work: On the iteration of quaternionic moebius transformations
full work available at URL: s0893-9659(02)80003-9
OpenAlex ID: W2095155041
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description: scientific article; zbMATH DE number 1781915