Technical Reports (CIS)
Title
Logarithms and Square Roots of Real Matrices
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.
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Date Posted: 24 April 2008
Comments
University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-08-12.