Matrix Logarithms

Consider the exponential map for matrices {\exp : \text{End}(\mathbb{C}^{m}) \rightarrow \text{Gl}(m, \mathbb{C})}, which can be defined by the formula

\displaystyle e^{x} = \sum_{k = 0}^{\infty} \frac{(2 \pi i)^k}{k!} x^{k},

where {x} is a square matrix. A matrix logarithm is defined to be a right-inverse to the exponential; that is to say, it is a function {\log: \text{Gl}(m, \mathbb{C}) \rightarrow \text{End}(\mathbb{C}^{m})} such that {\exp \circ \log = \text{id}_{ \text{Gl}(m, \mathbb{C})}}. It is a standard result of Lie theory that the exponential is a diffeomorphism in a neighbourhood of {0} in {\text{End}(\mathbb{C}^{m})}, and hence there is a well-defined choice of logarithm in a neighbourhood of the identity in {\text{Gl}(m, \mathbb{C})}. However, it is not at all clear that a logarithm can be globally defined, or that there is a unique way of doing so. This problem also shows up in the more simple case of the exponential for complex numbers. Indeed, there are several choices for the complex logarithm, and none of them are analytic (or even continuous) on all of {\mathbb{C}^{\times}}. However, it turns out that choosing a complex logarithm is enough to determine a unique choice of a matrix logarithm in any dimension. More precisely, we have the following:

Theorem. Consider the exponential sequence for complex numbers

\displaystyle 0 \rightarrow \mathbb{Z} \rightarrow \mathbb{C} \mathop{\rightarrow}^{\exp} \mathbb{C}^{\times} \rightarrow 0,

and let {\tau : \mathbb{C}^{\times} \rightarrow \mathbb{C}} be a set-theoretical splitting of this sequence, in other words, a (not-necessarily continuous) branch of the complex logarithm. Then for any positive integer {m}, there is a uniquely defined matrix logarithm

\displaystyle \log: \text{Gl}(m, \mathbb{C}) \rightarrow \text{End}(\mathbb{C}^{m})

such that for every {x \in \text{Gl}(m, \mathbb{C})} the eigenvalues of {\log(x)} lie in the image of {\tau.}

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