Square roots of \(3\times 3\) matrices
type: Item
instance of: scholarly article
title: Square roots of \(3\times 3\) matrices
zbMATH Open document ID: 1378.65096
author: Mehdi Ashkartizabi
author: Mina Aminghafari
author: Adel Mohammadpour
published in: Calcolo
publication date: 2017-10-27T00:00:00Z
review text: Square roots \(B\) of a matrix \(A\in\mathbb{C}^{n\times n}\) are computed by solving \(B^2=A\). First a new proof is given for the solution when \(n=2\), as proposed in an exercise in [\textit{N. J. Higham}, Functions of matrices. Theory and computation. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (2008; Zbl 1167.15001)]. In the main result, a similar strategy is used for \(n=3\). It states that if \(\det(A-MI)\neq0\), then \(B=(tA+\sqrt{\Delta}I)(A-MI)^{-1}\) where \(\Delta=\det(A)\), \(t=\mathrm{trace}(B)\), and \(M=0.5(T-t^2)\), with \(T=\mathrm{trace}(A)\). Since \(t\) satisfies a polynomial equation of degree 4 with a possible choice of a square root sign in a coefficient, there are generically up to 8 square roots. If \(M\) is an eigenvalue of \(A\), then \(\det(A-MI)=0\) and no square roots or coinciding square roots may occur. Matlab code is provided in an appendix to compute square roots of \(2\times 2\) and \(3\times 3\) matrices. It is illustrated numerically that the accuracy and computing time of the method is better than the results obtained by the standard matlab routine \texttt{sqrtm(\ )}.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 65F60
Mathematics Subject Classification ID: 15A12
Mathematics Subject Classification ID: 15A16
Mathematics Subject Classification ID: 65Y15
zbMATH DE Number: 6798976
zbMATH Keywords: Schur algorithm
zbMATH Keywords: square roots
zbMATH Keywords: Cayley-Hamilton theorem
zbMATH Keywords: numerical examples
zbMATH Keywords: Matlab code
describes a project that uses: mftoolbox
describes a project that uses: Matlab
MaRDI profile type: Publication
full work available at URL: s10092-016-0202-3
OpenAlex ID: W2509911834
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DOI: 10.1007/S10092-016-0202-3
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description: scientific article; zbMATH DE number 6798976