Extraction of nth roots of 2×2 matrices

Abstract

This paper is concerned with the determination of algebraic formulae giving all the solutions of the matrix equation X n = A where n is a positive integer greater than 2 and A is a 2×2 matrix with real or complex elements. If A is a 2×2 scalar matrix, the equation X n = A has infinitely many solutions and we obtain explicit formulae giving all the solutions. If A is a non-scalar 2×2 matrix, the equation X n = A has a finite number of solutions and we give a formula expressing all solutions in terms of A and the roots of a suitably defined nth 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, we get explicit algebraic formulae for all the solutions. We also determine the precise number of solutions in various cases.