Logarithms and Square Roots of Real Matrices

The need for computing logarithms or square roots of real matrices arises in a number of applied problems. A significant class of problems comes from medical imaging. One of these problems is to interpolate and to perform statistics on data represented by certain kinds of matrices (such as symmetric positive definite matrices in DTI). Another important and difficult problem is the registration of medical images. For both of these problems, the ability to compute logarithms of real matrices turns out to be crucial. However, not all real matrices have a real logarithm and thus, it is important to have sufficient conditions for the existence (and possibly the uniqueness) of a real logarithm for a real matrix. Such conditions (involving the eigenvalues of a matrix) are known, both for the logarithm and thesquare root.As far as I know, with the exception of Higham\u27s recent book [18], proofs of the results involving these conditions are scattered in the literature and it is not easy to locate them. Moreover, Higham\u27s excellent book assumes a certain level of background in linear algebra that readers interested in applications to medical imaging may not possess so we feel thata more elementary presentation might be a valuable supplement to Higham [18]. In this paper, I present a unified exposition of these results, including a proof of the existence of the Real Jordan Form, and give more direct proofs of some of these results using the RealJordan Form