Square roots of \(3\times 3\) matrices
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(\ )}.
scientific article; zbMATH DE number 6798976
scientific article