On square-rooting matrices

Following on from Nick MacKinnon’s work in the June 1989 Gazette concerning four different methods of square-rooting 2 × 2 matrices, we may ask if it is possible to compute all the square roots of any given 2 × 2 matrix, and if so, how? From now on by a “matrix” we mean a real or complex 2 ×2 matrix. We know that every non-zero number has precisely two square roots, but the situation is rather more complicated for matrices. In fact we shall see that the zero matrix and any other multiple of the identity matrix has an infinite number of square roots. All other matrices have only a finite number of square roots, with some (rather surprisingly) having none at all! Our method of enquiry is based on Nick MacKinnon's first method, namely matrix diagonalisation.