Some calculation methods of square root of a 2×2 matrix

Matrices have started to be used in computer and engineering technology to a great extent with affine transformations. In particular, 2x2 matrices are important for the study of planar motions. Finding the roots of matrices and especially 2x2 matrices is one of the major problems of linear algebra with geometry. In the literature, the square roots of 2x2 matrices are generally based on Schur's theorem, Cayley Hamilton's theorem and Newton's theorem. It can also be found with the help of sequence recurrence relations. In this study, first of all, some of the methods found in the literature will be given and their usefulness will be examined. In addition, it will be shown that the roots of some special matrices can be found with the help of complex numbers, dual numbers and hyperbolic numbers. Finally, how to find the roots of a matrix with the help of hybrid numbers will be examined in detail. In particular, how to find the square root of a matrix with the help of the idempotent base of hyperbolic hybrid numbers is a method not found in the literature. This method will be given in this thesis.