Consider the exponential map for matrices , which can be defined by the formula
where 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 such that . It is a standard result of Lie theory that the exponential is a diffeomorphism in a neighbourhood of in , and hence there is a well-defined choice of logarithm in a neighbourhood of the identity in . 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 . 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
and let 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 , there is a uniquely defined matrix logarithm
such that for every the eigenvalues of lie in the image of
1. Matrix decompositions
Before discussing matrix logarithms, we’ll start by stating two decomposition results for square matrices. Recall that a matrix is semi-simple if it is diagonalizable, and that it is nilpotent if it gives when raised to some positive power. The Jordan-Chevalley decomposition can be stated as follows:
Proposition. Given a square matrix , there are unique matrices , such that is semi-simple, is nilpotent, and commute, and . Furthermore, and commute with any matrix that commutes with . Note also that and have the same set of eigenvalues.
The proof for this is standard and can be found, for example, in Humphreys’ book on Lie algebras and representation theory. Existence of the decomposition is obvious from Jordan normal form.
A corollary to this is a multiplicative version of the decomposition theorem in the case of invertible matrices. Recall that a matrix is unipotent if is nilpotent.
Corollary. Given an invertible matrix , there are unique invertible matrices and such that is semi-simple, is unipotent, and commute, and . Furthermore, and commute with any matrix which commutes with . Note also that and have the same set of eigenvalues.
Sketch of proof. Using the Jordan-Chevalley decomposition we get for semi-simple and nilpotent. Then ; so let , . Thus , for semi-simple, and unipotent.
2. Matrix logarithm for Unipotent matrices
We begin by describing the matrix exponential and logarithm in the case of nilpotent and unipotent matrices. Let denote the subspace of nilpotent matrices, and let denote the subset of unipotent matrices. If is nilpotent, then observe that
is a finite sum and is therefore unipotent. Hence the matrix exponential gives a map . On the other hand, given a unipotent matrix , the formal series expansion for the logarithm
is a finite sum of commuting nilpotent matrices, and is therefore a nilpotent matrix. Hence we have a map which sends unipotent matrices to nilpotent matrices: . Since the series expansions for and are formally inverse to each other, our definitions for and in the present case are actually inverse to each other. Hence indeed defines a matrix logarithm and we have the following proposition
Proposition The exponential gives a bijection between the nilpotent matrices and the unipotent matrices. In particular, given a unipotent matrix , there is a unique nilpotent logarithm .
3. Matrix logarithm for Semi-simple matrices
We now describe the matrix logarithm in the case of semi-simple matrices. We will see that in this case, in order to define a matrix logarithm, we must first choose a set theoretical splitting of the exponential sequence as above. We begin with the following preliminary fact:
Lemma. is semi-simple if and only if is semi-simple.
Proof. One direction is obvious: if is semi-simple, then for diagonal, and hence , where is diagonal. In general, let ; in this case . Since is nilpotent,
is also nilpotent. Furthermore, since and commute, is the Jordan-Chevalley decomposition. Hence, by uniqueness of this decomposition, if is semi-simple, then and . But this implies that
Choose a basis such that is strictly upper triangular. If , let be a nonzero entry such that is maximal. Then for , . Hence , a contradiction. Hence , and is semi-simple.
Corollary. Given any choice of a matrix logarithm, we have that is semi-simple if and only if is semi-simple.
We can now define a matrix logarithm in a unique way for semi-simple matrices given a choice of splitting :
Theorem. Given a splitting of the exponential sequence as above, and a semi-simple invertible matrix , there is a unique logarithm whose eigenvalues lie in the image of .
Proof. We begin with existence: is semi-simple, so , for diagonal matrix
where all . Define
Then solves the problem.
Uniqueness: Suppose is a logarith of such that its eigenvalues lie in the image of . Then since is semi-simple, so is and hence for diagonal matrix (with diagonal entries in the image of ). Hence
The diagonal matrices and are conjugate, and therefore differ only by reordering of entries: there is a permutation matrix such that . Then , with . So there is no loss in generality by assuming that with . But then we have that for all . And since the eigenvalues of lie in the image of , we have that . Hence , implying that . But then , so . And since we have that , it follows that , thus proving uniqueness.
4. Matrix logarithm for invertible matrices
By combining the results above for unipotent and semi-simple matrices, we will show that we get a uniquely defined matrix logarithm for any invertible matrix, once we have chosen a set-theoretical splitting of the exponential sequence. Begin by choosing such a splitting , thus fixing a logarithm for semi-simple matrices. We begin with the following fact:
Lemma. Let be a semi-simple invertible matrix, and let be a matrix which commutes with . Then also commutes with .
Proof. Consider the decomposition of into the eigenspaces of :
where for . Since commutes with , it preserves the above direct sum decomposition. The logarithm of is such that given , . So given a vector , which can be written according to the direct sum decomposition as , we have
Theorem. Given a splitting of the exponential sequence as above, and an invertible matrix , there is a unique logarithm whose eigenvalues lie in the image of .
Proof. We begin with existence. The multiplicative Jordan-Chevalley decomposition lets us uniquely write , with semi-simple and unipotent, and with and commuting. Therefore, both and can be defined as above. By the previous lemma, commutes with , and therefore, with
Therefore, define , which is already expressed in its additive Jordan-Chevalley decomposition. That this actually defines a logarithm can be seen as follows:
where we have used the fact that and commute. Furthermore, since the eigenvalues of are eigenvalues of , they all lie in the image of . This shows existence.
Now for uniqueness, suppose we have a logarithm of , whose eigenvalues lie in the image of . Let be the Jordan-Chevalley decomposition. Note that the eigenvalues of also lie in the image of . Then we have
Since is semi-simple, is unipotent, and the matrices commute, this must be the multiplicative Jordan-Chevalley decomposition of : hence and . So is a logarithm of whose eigenvalues lie in the image of ; hence . And . Therefore, , proving uniqueness.
Remark. The matrix exponential sends the additive Jordan-Chevalley decomposition to the multiplicative one, and the matrix logarithm sends the multiplicative decomposition to the additive one. As such, we can view the matrix exponential and logarithm as acting component-wise on the Jordan-Chevalley decompositions.
We know that for commuting matrices and , their exponentials also commute. The same is true for the matrix logarithm.
Corollary. If and commute, then and commute.