Extraction of \(n\)th roots of \(2 \times 2\) matrices
Let \(A\) be a square matrix whose elements are either real or complex numbers. Any matrix \(X\) such that \(X^n=A\) is called an \(n\)th root of \(A\) and the problem of determining all the \(n\)th roots of a given matrix is classic, but explicit formulae giving \(X\) in terms of \(A\) and its elements are not generally known. \textit{A. Cayley} [A memoir on the theory of matrices, Phil. Trans. Roy. Soc. London 148, 17-37 (1858)] and \textit{D. Sullivan} [Math. Mag. 66, No. 5, 314--316 (1993; Zbl 0803.15012)] have obtained algebraic formulae giving the square roots of \(2{\times}2\) matrices and \textit{P. Damphousse} [Fibonacci Q. 27, No. 5, 386--401 (1989; Zbl 0679.10012)] has given formulae expressing the \(n\)th roots of non-singular \(2{\times}2\) matrices in terms of transcendental functions. In the paper under review the author considers formulae giving all the solutions of the matrix equation \(X^n=A\) where \(n \geq3\) and \(A\) is a \(2{\times}2\) matrix. He first obtains explicit algebraic formulae given the infinite set of all \(n\)th roots of scalar \(2{\times}2\) matrices. If \(A\) is a non-scalar \(2{\times}2\) matrix, the equation \(X^n=A\) has a finite number of solutions and the author gives a formula expressing all solutions in terms of \(A\) and the roots of a suitably defined \(n\)th degree polynomial in a single variable. This leads to very simple formulae for all the solutions when \(A\) is either a singular matrix or a non-singular matrix with two coincident eigenvalues. Similarly when \(n=3\) or \(4\), he presents explicit algebraic formulae for all the solutions. Moreover he also determines the precise number of solutions in various cases.
\(n\)th roots of \(2{\times}2\) matrices
scientific article; zbMATH DE number 2081244