Technical Reports (CIS)
Document Type
Technical Report
Date of this Version
May 2008
Abstract
In these notes, we consider the problem of finding the logarithm or the square root of a real matrix. It is known that for every real n × n matrix, A, if no real eigenvalue of A is negative or zero, then A has a real logarithm, that is, there is a real matrix, X, such that eX = A. Furthermore, if the eigenvalues, ξ, of X satisfy the property −π < ℑ(ξ) < π, then X is unique. It is also known that under the same condition every real n × n matrix, A, has a real square root, that is, there is a real matrix, X, such that X2 = A. Moreover, if the eigenvalues, ρeἰθ, of X satisfy the condition −π/ 2 < θ < π/ 2, then X is unique. These theorems are the theoretical basis for various numerical methods for exponentiating a matrix or for computing its logarithm using a method known as scaling and squaring (resp. inverse scaling and squaring). Such methods play an important role in the log-Euclidean framework due to Arsigny, Fillard, Pennec and Ayache and its applications to medical imaging. Actually, there is a necessary and sufficient condition for a real matrix to have a real logarithm (or a real square root) but it is fairly subtle as it involves the parity of the number of Jordan blocks associated with negative eigenvalues. As far as I know, proofs of these results are scattered in the literature and it is not easy to locate them. In these notes, I present a unified exposition of these results and give more direct proofs of some of them using the Real Jordan Form.
Date Posted: 24 April 2008
Comments
University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-08-12.