Because of Euler’s identity , the complex exponential is not injective: for any complex and integer . As such, the complex logarithm is not well-defined as a single-valued function from to . However, after making a branch cut, one can create a branch of the logarithm which is single-valued. For instance, after removing the negative real axis , one has the *standard branch* of the logarithm, with defined as the unique choice of the complex logarithm of whose imaginary part has magnitude strictly less than . This particular branch has a number of useful additional properties:

- The standard branch is holomorphic on its domain .
- One has for all in the domain . In particular, if is real, then is real.
- One has for all in the domain .

One can then also use the standard branch of the logarithm to create standard branches of other multi-valued functions, for instance creating a standard branch of the square root function. We caution however that the identity can fail for the standard branch (or indeed for any branch of the logarithm).

One can extend this standard branch of the logarithm to complex matrices, or (equivalently) to linear transformations on an -dimensional complex vector space , provided that the spectrum of that matrix or transformation avoids the branch cut . Indeed, from the spectral theorem one can decompose any such as the direct sum of operators on the non-trivial generalised eigenspaces of , where ranges in the spectrum of . For each component of , we define

where is the Taylor expansion of at ; as is nilpotent, only finitely many terms in this Taylor expansion are required. The logarithm is then defined as the direct sum of the .

The matrix standard branch of the logarithm has many pleasant and easily verified properties (often inherited from their scalar counterparts), whenever has no spectrum in :

- (i) We have .
- (ii) If and have no spectrum in , then .
- (iii) If has spectrum in a closed disk in , then , where is the Taylor series of around (which is absolutely convergent in ).
- (iv) depends holomorphically on . (Easily established from (ii), (iii), after covering the spectrum of by disjoint disks; alternatively, one can use the Cauchy integral representation for a contour in the domain enclosing the spectrum of .) In particular, the standard branch of the matrix logarithm is smooth.
- (v) If is any invertible linear or antilinear map, then . In particular, the standard branch of the logarithm commutes with matrix conjugations; and if is real with respect to a complex conjugation operation on (that is to say, an antilinear involution), then is real also.
- (vi) If denotes the transpose of (with the complex dual of ), then . Similarly, if denotes the adjoint of (with the complex conjugate of , i.e. with the conjugated multiplication map ), then .
- (vii) One has .
- (viii) If denotes the spectrum of , then .

As a quick application of the standard branch of the matrix logarithm, we have

Proposition 1Let be one of the following matrix groups: , , , , , or , where is a non-degenerate real quadratic form (so is isomorphic to a (possibly indefinite) orthogonal group for some . Then any element of whose spectrum avoids is exponential, that is to say for some in the Lie algebra of .

*Proof:* We just prove this for , as the other cases are similar (or a bit simpler). If , then (viewing as a complex-linear map on , and using the complex bilinear form associated to to identify with its complex dual , then is real and . By the properties (v), (vi), (vii) of the standard branch of the matrix logarithm, we conclude that is real and , and so lies in the Lie algebra , and the claim now follows from (i).

Exercise 2Show that is not exponential in if . Thus we see that the branch cut in the above proposition is largely necessary. See this paper of Djokovic for a more complete description of the image of the exponential map in classical groups, as well as this previous blog post for some more discussion of the surjectivity (or lack thereof) of the exponential map in Lie groups.

For a slightly less quick application of the standard branch, we have the following result (recently worked out in the answers to this MathOverflow question):

Proposition 3Let be an element of the split orthogonal group which lies in the connected component of the identity. Then .

The requirement that lie in the identity component is necessary, as the counterexample for shows.

*Proof:* We think of as a (real) linear transformation on , and write for the quadratic form associated to , so that . We can split , where is the sum of all the generalised eigenspaces corresponding to eigenvalues in , and is the sum of all the remaining eigenspaces. Since and are real, are real (i.e. complex-conjugation invariant) also. For , the restriction of to then lies in , where is the restriction of to , and

The spectrum of consists of positive reals, as well as complex pairs (with equal multiplicity), so . From the preceding proposition we have for some ; this will be important later.

It remains to show that . If has spectrum at then we are done, so we may assume that has spectrum only at (being invertible, has no spectrum at ). We split , where correspond to the portions of the spectrum in , ; these are real, -invariant spaces. We observe that if are generalised eigenspaces of with , then are orthogonal with respect to the (complex-bilinear) inner product associated with ; this is easiest to see first for the actual eigenspaces (since for all ), and the extension to generalised eigenvectors then follows from a routine induction. From this we see that is orthogonal to , and and are null spaces, which by the non-degeneracy of (and hence of the restriction of to ) forces to have the same dimension as , indeed now gives an identification of with . If we let be the restrictions of to , we thus identify with , since lies in ; in particular is invertible. Thus

and so it suffices to show that .

At this point we need to use the hypothesis that lies in the identity component of . This implies (by a continuity argument) that the restriction of to any maximal-dimensional positive subspace has positive determinant (since such a restriction cannot be singular, as this would mean that positive norm vector would map to a non-positive norm vector). Now, as have equal dimension, has a balanced signature, so does also. Since , already lies in the identity component of , and so has positive determinant on any maximal-dimensional positive subspace of . We conclude that has positive determinant on any maximal-dimensional positive subspace of .

We choose a complex basis of , to identify with , which has already been identified with . (In coordinates, are now both of the form , and for .) Then becomes a maximal positive subspace of , and the restriction of to this subspace is conjugate to , so that

But since and is positive definite, so as required.

## 24 comments

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4 May, 2015 at 1:36 am

AnonymousIt seems that the standard logarithm can be extended for a more general class of operators on a complex Banach space via its Cauchy integral formula representation (provided that the integration contour encloses the spectrum of and avoids the branch cut).

[Remark added – T.]4 May, 2015 at 8:26 pm

AnonymousIn the Cauchy integral representation (property (iv) before proposition 1), the denominator in the integrand should be (but it seems better to use the standard notation for the nonscalar resolvent of ).

[Corrected, thanks – T.]4 May, 2015 at 5:30 am

Leo SteinYou currently have ${z\in \mathbb{C} : |\Im z| < \pi }$ in your definition of the standard branch of Log. I believe this is a typo, and you mean \arg instead of \Im

[The restriction on is on the range of Log, not the domain – T.]4 May, 2015 at 5:43 am

Jarred BarberYou have a minor typo in Prop 1: 0 <= k <= n-k should be 0 <= k <= n

[Corrected – T.]4 May, 2015 at 6:01 am

DanielIt may be good to remark that the standard branch (like any branch) of the logarithm also has one not so nice property: In general Log (zw) is not equal to Log z + Log w. For the multivalued function log it is true that log (zw) = log z + log w in the sense of (Minkowski) sum of sets.

[Remark added – T.]4 May, 2015 at 8:27 am

AnonymousThe identity is satisfied for the standard branch near in and therefore (by analytic continuation) also globally on the product of the corresponding domains (for and ) in .

4 May, 2015 at 10:17 am

AnonymousCorrection: In order to use the analytic continuation argument, the domains of and should be sufficiently restricted (e.g. to the open right half plane) so that their “productset” should be connected and not intersecting the branch line (i.e. should be well defined).

4 May, 2015 at 6:47 am

AnonymousSince the standard logarithm is holomorphic on its domain, it is uniquely determined (via analytic continuation) by its value at any single point in the domain (e.g. at ).

4 May, 2015 at 1:16 pm

BenTypo: you wrote Log(zw)=Log(z)Log(w).

[Corrected, thanks – T.]4 May, 2015 at 3:55 pm

David SpeyerThis choice of standard branch has the slightly odd property that, if is a Lie subgroup of , then may not land in the Lie algebra of . For example, if is , and has eigenvalues , and , then your logarithm will have eigenvalues and hence trace , not . Indeed, for , I think your fundamental domain is disconnected, having three regions where the logarithm has trace , and

A while back I noticed a way to fix this for simply connected compact groups. Let be the Lie algebra. Let be the set of such that the spectrum of lies in . Then is a convex neighborhood of the origin in , and the exponential map is an open immersion with dense image. In the case, this means that we lift in the unique manner such that , and and . For example, we lift to .

The argument I found went through a bunch of root system combinatorics, although I assume there is a simple explanation for it that I missed. I'd be curious whether people have seen this before.

4 May, 2015 at 7:08 pm

Allen KnutsonFor a simply connected compact group, consider the composite where is the coweight lattice and the Weyl alcove. Now consider the preimage of the far wall of the Weyl alcove; I think your is the connected component around of the complement.

Maybe this is the root system combinatorics you were talking about. Anyway yes, at least one person has seen it before.

5 May, 2015 at 6:52 am

David SpeyerYup, that’s the root system combinatorics I was thinking about, although I phrased it differently in my mind.

5 May, 2015 at 10:55 pm

Lei WangAbout Proposition 3, what about lies in the other components of ? I feel one can still show bounds on .

6 May, 2015 at 7:49 am

Terence TaoIt appears that the same argument gives either or for any , where the sign of the inequality depends on whether the restriction (and projection) of to a maximal totally positive subspace is orientation preserving or not.

6 May, 2015 at 8:04 am

Lei WangThere are four components in total, right ? Can one prove a stronger statement like in one (or even two) of the four ?

6 May, 2015 at 8:08 am

Terence TaoI believe this is the case when is odd. For instance, for odd , the component of containing will necessarily have an odd number of eigenvalues at -1 (counted with algebraic multiplicity), since all nearby eigenvalues come in pairs , and so is necessarily zero on this component. (One way to think about this is that one inequality comes from considering the projection of onto the totally positive space, and the other inequality comes from analysing the totally negative space. When is even, it looks like both arguments give the same inequality rather than opposing inequalities.)

6 May, 2015 at 1:56 pm

Lei WangIt appears to me this is independent of . In the and component one has , while in the component ; component . Again, just an empirical observation plus some guesswork from a computational physicists :-)

6 May, 2015 at 2:30 pm

Terence TaoI think you are correct. In particular, is always singular on the components. It would be interesting to have an elementary proof of this in the case when n is even (the argument I gave only worked then n is odd). It generalises the assertion that an orthogonal matrix of determinant -1 necessarily has an odd number of eigenvalues at -1 (because the product of all the other eigenvalues is positive), regardless of the parity of dimension.

2 June, 2015 at 7:23 am

Gergely HarcosDear Lei, here is a direct argument that you asked for in our email correspondence. Assume that . Following the blog entry, we can assume that has no spectrum at , for otherwise . Then,

,

hence . Furthermore, the sign of equals the sign of , which in turn equals depending whether lies in the component or the component. Done.

9 May, 2015 at 9:10 am

AnonymousI share the assessment of:

therefore ; we have.

$latex e^{\frac{1+i \sqrt{n}}{n}} = e^{\frac{1}{n}}

15 May, 2015 at 3:28 am

Sign problems, Terry Tau, and open science | This Condensed Life[…] For spin unpolarized systems it’s pretty clear why each A_i is of this form, but for others systems it’s not as clear. Anyways, this was the question he posed to mathoverflow. Amazingly people jumped on it, and it wasn’t long before Fields Medalist Terrance Tao got involved! It was then only a matter of 3 days before the conjecture was proven to be true in a mathematically rigorous way. For a full writeup of the proof, you can check out Tao’s blog. […]

31 May, 2015 at 9:48 am

E.L. WistyReblogged this on Pink Iguana.

26 September, 2015 at 6:09 am

gninrepoliThere is a clear link between the natural logarithm, and trigonometric functions (cosine, sine, which are defined on the two-dimensional space). I have a question. Built on whether the model such as four-dimensional space? I mean the same functions for the study of the nature of prime numbers for example.Until I see the study of the same prime numbers only with a two-dimensional space.And I’m still interested in the nature of prime numbers in what space they are located? Maybe in the third? At least describe them by using the same two-dimensional space is difficult. Perhaps this is more a philosophy than mathematics.

26 September, 2015 at 6:25 am

gninrepoliI mean the opening of a Cartesian system, and then the natural logarithm. Later there were complex numbers.