The standard branch of the matrix logarithm

Because of Euler’s identity Image may be NSFW.
Clik here to view. ${e^{\pi i} + 1 = 0}$ , the complex exponential is not injective: Image may be NSFW.
Clik here to view. ${e^{z + 2\pi i k} = e^z}$ for any complex Image may be NSFW.
Clik here to view. {z} and integer Image may be NSFW.
Clik here to view. {k} . As such, the complex logarithm Image may be NSFW.
Clik here to view. ${z \mapsto \log z}$ is not well-defined as a single-valued function from Image may be NSFW.
Clik here to view. ${{\bf C} \backslash \{0\}}$ to Image may be NSFW.
Clik here to view. ${{\bf C}}$ . 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 Image may be NSFW.
Clik here to view. ${(-\infty,0]}$ , one has the standard branch Image may be NSFW.
Clik here to view. ${\hbox{Log}: {\bf C} \backslash (-\infty,0] \rightarrow \{ z \in {\bf C}: |\hbox{Im} z| < \pi \}}$ of the logarithm, with Image may be NSFW.
Clik here to view. ${\hbox{Log}(z)}$ defined as the unique choice of the complex logarithm of Image may be NSFW.
Clik here to view. {z} whose imaginary part has magnitude strictly less than Image may be NSFW.
Clik here to view. ${\pi}$ . This particular branch has a number of useful additional properties:

The standard branch Image may be NSFW.
Clik here to view. ${\hbox{Log}}$ is holomorphic on its domain Image may be NSFW.
Clik here to view. ${{\bf C} \backslash (-\infty,0]}$ .
One has Image may be NSFW.
Clik here to view. ${\hbox{Log}( \overline{z} ) = \overline{ \hbox{Log}(z) }}$ for all Image may be NSFW.
Clik here to view. in the domain Image may be NSFW.
Clik here to view. ${{\bf C} \backslash (-\infty,0]}$ . In particular, if Image may be NSFW.
Clik here to view. ${z \in {\bf C} \backslash (-\infty,0]}$ is real, then Image may be NSFW.
Clik here to view. ${\hbox{Log} z}$ is real.
One has Image may be NSFW.
Clik here to view. ${\hbox{Log}( z^{-1} ) = - \hbox{Log}(z)}$ for all Image may be NSFW.
Clik here to view. in the domain Image may be NSFW.
Clik here to view. ${{\bf C} \backslash (-\infty,0]}$ .

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 Image may be NSFW.
Clik here to view. ${z \mapsto \exp( \frac{1}{2} \hbox{Log} z )}$ of the square root function. We caution however that the identity Image may be NSFW.
Clik here to view. ${\hbox{Log}(zw) = \hbox{Log}(z) + \hbox{Log}(w)}$ can fail for the standard branch (or indeed for any branch of the logarithm).

One can extend this standard branch of the logarithm to Image may be NSFW.
Clik here to view. ${n \times n}$ complex matrices, or (equivalently) to linear transformations Image may be NSFW.
Clik here to view. ${T: V \rightarrow V}$ on an Image may be NSFW.
Clik here to view. {n} -dimensional complex vector space Image may be NSFW.
Clik here to view. {V} , provided that the spectrum of that matrix or transformation avoids the branch cut Image may be NSFW.
Clik here to view. ${(-\infty,0]}$ . Indeed, from the spectral theorem one can decompose any such Image may be NSFW.
Clik here to view. ${T: V \rightarrow V}$ as the direct sum of operators Image may be NSFW.
Clik here to view. ${T_\lambda: V_\lambda \rightarrow V_\lambda}$ on the non-trivial generalised eigenspaces Image may be NSFW.
Clik here to view. ${V_\lambda}$ of Image may be NSFW.
Clik here to view. {T} , where Image may be NSFW.
Clik here to view. ${\lambda \in {\bf C} \backslash (-\infty,0]}$ ranges in the spectrum of Image may be NSFW.
Clik here to view. {T} . For each component Image may be NSFW.
Clik here to view. ${T_\lambda}$ of Image may be NSFW.
Clik here to view. {T} , we define

Image may be NSFW.
Clik here to view. $\displaystyle \hbox{Log}( T_\lambda ) = P_\lambda( T_\lambda )$

where Image may be NSFW.
Clik here to view. ${P_\lambda}$ is the Taylor expansion of Image may be NSFW.
Clik here to view. ${\hbox{Log}}$ at Image may be NSFW.
Clik here to view. ${\lambda}$ ; as Image may be NSFW.
Clik here to view. ${T_\lambda-\lambda}$ is nilpotent, only finitely many terms in this Taylor expansion are required. The logarithm Image may be NSFW.
Clik here to view. ${\hbox{Log} T}$ is then defined as the direct sum of the Image may be NSFW.
Clik here to view. ${\hbox{Log} T_\lambda}$ .

The matrix standard branch of the logarithm has many pleasant and easily verified properties (often inherited from their scalar counterparts), whenever Image may be NSFW.
Clik here to view. ${T: V \rightarrow V}$ has no spectrum in Image may be NSFW.
Clik here to view. ${(-\infty,0]}$ :

(i) We have Image may be NSFW.
Clik here to view. ${\exp( \hbox{Log} T ) = T}$ .
(ii) If Image may be NSFW.
Clik here to view. ${T_1: V_1 \rightarrow V_1}$ and Image may be NSFW.
Clik here to view. ${T_2: V_2 \rightarrow V_2}$ have no spectrum in Image may be NSFW.
Clik here to view. ${(-\infty,0]}$ , then Image may be NSFW.
Clik here to view. ${\hbox{Log}( T_1 \oplus T_2 ) = \hbox{Log}(T_1) \oplus \hbox{Log}(T_2)}$ .
(iii) If Image may be NSFW.
Clik here to view. has spectrum in a closed disk Image may be NSFW.
Clik here to view. in Image may be NSFW.
Clik here to view. ${{\bf C} \backslash (-\infty,0]}$ , then Image may be NSFW.
Clik here to view. ${\hbox{Log}(T) = P_z(T)}$ , where Image may be NSFW.
Clik here to view. ${P_z}$ is the Taylor series of Image may be NSFW.
Clik here to view. ${\hbox{Log}}$ around Image may be NSFW.
Clik here to view. (which is absolutely convergent in Image may be NSFW.
Clik here to view.).
(iv) Image may be NSFW.
Clik here to view. ${\hbox{Log}(T)}$ depends holomorphically on Image may be NSFW.
Clik here to view.. (Easily established from (ii), (iii), after covering the spectrum of Image may be NSFW.
Clik here to view. by disjoint disks; alternatively, one can use the Cauchy integral representation Image may be NSFW.
Clik here to view. ${\hbox{Log}(T) = \frac{1}{2\pi i} \int_\gamma \hbox{Log}(z)(z-T)^{-1}\ dz}$ for a contour Image may be NSFW.
Clik here to view. ${\gamma}$ in the domain enclosing the spectrum of Image may be NSFW.
Clik here to view..) In particular, the standard branch of the matrix logarithm is smooth.
(v) If Image may be NSFW.
Clik here to view. ${S: V \rightarrow W}$ is any invertible linear or antilinear map, then Image may be NSFW.
Clik here to view. ${\hbox{Log}( STS^{-1} ) = S \hbox{Log}(T) S^{-1}}$ . In particular, the standard branch of the logarithm commutes with matrix conjugations; and if Image may be NSFW.
Clik here to view. is real with respect to a complex conjugation operation on Image may be NSFW.
Clik here to view. (that is to say, an antilinear involution), then Image may be NSFW.
Clik here to view. ${\hbox{Log}(T)}$ is real also.
(vi) If Image may be NSFW.
Clik here to view. ${T^*: V^* \rightarrow V^*}$ denotes the transpose of Image may be NSFW.
Clik here to view. (with Image may be NSFW.
Clik here to view. ${V^*}$ the complex dual of Image may be NSFW.
Clik here to view.), then Image may be NSFW.
Clik here to view. ${\hbox{Log}(T^*) = \hbox{Log}(T)^*}$ . Similarly, if Image may be NSFW.
Clik here to view. ${T^\dagger: V^\dagger \rightarrow V^\dagger}$ denotes the adjoint of Image may be NSFW.
Clik here to view. (with Image may be NSFW.
Clik here to view. ${V^\dagger}$ the complex conjugate of Image may be NSFW.
Clik here to view. ${V^*}$ , i.e. Image may be NSFW.
Clik here to view. ${V^*}$ with the conjugated multiplication map Image may be NSFW.
Clik here to view. ${(c,z) \mapsto \overline{c} z}$ ), then Image may be NSFW.
Clik here to view. ${\hbox{Log}(T^\dagger) = \hbox{Log}(T)^\dagger}$ .
(vii) One has Image may be NSFW.
Clik here to view. ${\hbox{Log}(T^{-1}) = - \hbox{Log}( T )}$ .
(viii) If Image may be NSFW.
Clik here to view. ${\sigma(T)}$ denotes the spectrum of Image may be NSFW.
Clik here to view., then Image may be NSFW.
Clik here to view. ${\sigma(\hbox{Log} T) = \hbox{Log} \sigma(T)}$ .

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

Proposition 1 Let Image may be NSFW.
Clik here to view. be one of the following matrix groups: Image may be NSFW.
Clik here to view. ${GL_n({\bf C})}$ , Image may be NSFW.
Clik here to view. ${GL_n({\bf R})}$ , Image may be NSFW.
Clik here to view. ${U_n({\bf C})}$ , Image may be NSFW.
Clik here to view., Image may be NSFW.
Clik here to view. ${Sp_{2n}({\bf C})}$ , or Image may be NSFW.
Clik here to view. ${Sp_{2n}({\bf R})}$ , where Image may be NSFW.
Clik here to view. ${Q: {\bf R}^n \rightarrow {\bf R}}$ is a non-degenerate real quadratic form (so Image may be NSFW.
Clik here to view. is isomorphic to a (possibly indefinite) orthogonal group Image may be NSFW.
Clik here to view. for some Image may be NSFW.
Clik here to view. ${0 \leq k \leq n}$ . Then any element Image may be NSFW.
Clik here to view. of Image may be NSFW.
Clik here to view. whose spectrum avoids Image may be NSFW.
Clik here to view. ${(-\infty,0]}$ is exponential, that is to say Image may be NSFW.
Clik here to view. ${T = \exp(X)}$ for some Image may be NSFW.
Clik here to view. in the Lie algebra Image may be NSFW.
Clik here to view. ${{\mathfrak g}}$ of Image may be NSFW.
Clik here to view..

Proof: We just prove this for Image may be NSFW.
Clik here to view. {G=O(Q)} , as the other cases are similar (or a bit simpler). If Image may be NSFW.
Clik here to view. ${T \in O(Q)}$ , then (viewing Image may be NSFW.
Clik here to view. {T} as a complex-linear map on Image may be NSFW.
Clik here to view. ${{\bf C}^n}$ , and using the complex bilinear form associated to Image may be NSFW.
Clik here to view. {Q} to identify Image may be NSFW.
Clik here to view. ${{\bf C}^n}$ with its complex dual Image may be NSFW.
Clik here to view. ${({\bf C}^n)^*}$ , then Image may be NSFW.
Clik here to view. {T} is real and Image may be NSFW.
Clik here to view. ${T^{*-1} = T}$ . By the properties (v), (vi), (vii) of the standard branch of the matrix logarithm, we conclude that Image may be NSFW.
Clik here to view. ${\hbox{Log} T}$ is real and Image may be NSFW.
Clik here to view. ${- \hbox{Log}(T)^* = \hbox{Log}(T)}$ , and so Image may be NSFW.
Clik here to view. ${\hbox{Log}(T)}$ lies in the Lie algebra Image may be NSFW.
Clik here to view. ${{\mathfrak g} = {\mathfrak o}(Q)}$ , and the claim now follows from (i). Image may be NSFW.
Clik here to view. $\Box$

Exercise 2 Show that Image may be NSFW.
Clik here to view. ${\hbox{diag}(-\lambda, -1/\lambda)}$ is not exponential in Image may be NSFW.
Clik here to view. ${GL_2({\bf R})}$ if Image may be NSFW.
Clik here to view. ${-\lambda \in (-\infty,0) \backslash \{-1\}}$ . 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 3 Let Image may be NSFW.
Clik here to view. be an element of the split orthogonal group Image may be NSFW.
Clik here to view. which lies in the connected component of the identity. Then Image may be NSFW.
Clik here to view. ${\hbox{det}(1+T) \geq 0}$ .

The requirement that Image may be NSFW.
Clik here to view. {T} lie in the identity component is necessary, as the counterexample Image may be NSFW.
Clik here to view. ${T = \hbox{diag}(-\lambda, -1/\lambda )}$ for Image may be NSFW.
Clik here to view. ${\lambda \in (-\infty,-1) \cup (-1,0)}$ shows.

Proof: We think of Image may be NSFW.
Clik here to view. {T} as a (real) linear transformation on Image may be NSFW.
Clik here to view. ${{\bf C}^{2n}}$ , and write Image may be NSFW.
Clik here to view. {Q} for the quadratic form associated to Image may be NSFW.
Clik here to view. {O(n,n)} , so that Image may be NSFW.
Clik here to view. ${O(n,n) \equiv O(Q)}$ . We can split Image may be NSFW.
Clik here to view. ${{\bf C}^{2n} = V_1 \oplus V_2}$ , where Image may be NSFW.
Clik here to view. ${V_1}$ is the sum of all the generalised eigenspaces corresponding to eigenvalues in Image may be NSFW.
Clik here to view. ${(-\infty,0]}$ , and Image may be NSFW.
Clik here to view. ${V_2}$ is the sum of all the remaining eigenspaces. Since Image may be NSFW.
Clik here to view. {T} and Image may be NSFW.
Clik here to view. ${(-\infty,0]}$ are real, Image may be NSFW.
Clik here to view. ${V_1,V_2}$ are real (i.e. complex-conjugation invariant) also. For Image may be NSFW.
Clik here to view. {i=1,2} , the restriction Image may be NSFW.
Clik here to view. ${T_i: V_i \rightarrow V_i}$ of Image may be NSFW.
Clik here to view. {T} to Image may be NSFW.
Clik here to view. ${V_i}$ then lies in Image may be NSFW.
Clik here to view. ${O(Q_i)}$ , where Image may be NSFW.
Clik here to view. ${Q_i}$ is the restriction of Image may be NSFW.
Clik here to view. {Q} to Image may be NSFW.
Clik here to view. ${V_i}$ , and

Image may be NSFW.
Clik here to view. $\displaystyle \hbox{det}(1+T) = \hbox{det}(1+T_1) \hbox{det}(1+T_2).$

The spectrum of Image may be NSFW.
Clik here to view. ${T_2}$ consists of positive reals, as well as complex pairs Image may be NSFW.
Clik here to view. ${\lambda, \overline{\lambda}}$ (with equal multiplicity), so Image may be NSFW.
Clik here to view. ${\hbox{det}(1+T_2) > 0}$ . From the preceding proposition we have Image may be NSFW.
Clik here to view. ${T_2 = \exp( X_2 )}$ for some Image may be NSFW.
Clik here to view. ${X_2 \in {\mathfrak o}(Q_2)}$ ; this will be important later.

It remains to show that Image may be NSFW.
Clik here to view. ${\hbox{det}(1+T_1) \geq 0}$ . If Image may be NSFW.
Clik here to view. ${T_1}$ has spectrum at Image may be NSFW.
Clik here to view. {-1} then we are done, so we may assume that Image may be NSFW.
Clik here to view. ${T_1}$ has spectrum only at Image may be NSFW.
Clik here to view. ${(-\infty,-1) \cup (-1,0)}$ (being invertible, Image may be NSFW.
Clik here to view. {T} has no spectrum at Image may be NSFW.
Clik here to view. {0} ). We split Image may be NSFW.
Clik here to view. ${V_1 = V_3 \oplus V_4}$ , where Image may be NSFW.
Clik here to view. ${V_3,V_4}$ correspond to the portions of the spectrum in Image may be NSFW.
Clik here to view. ${(-\infty,-1)}$ , Image may be NSFW.
Clik here to view. {(-1,0)} ; these are real, Image may be NSFW.
Clik here to view. {T} -invariant spaces. We observe that if Image may be NSFW.
Clik here to view. ${V_\lambda, V_\mu}$ are generalised eigenspaces of Image may be NSFW.
Clik here to view. {T} with Image may be NSFW.
Clik here to view. ${\lambda \mu \neq 1}$ , then Image may be NSFW.
Clik here to view. ${V_\lambda, V_\mu}$ are orthogonal with respect to the (complex-bilinear) inner product Image may be NSFW.
Clik here to view. ${\cdot}$ associated with Image may be NSFW.
Clik here to view. {Q} ; this is easiest to see first for the actual eigenspaces (since Image may be NSFW.
Clik here to view. ${ \lambda \mu u \cdot v = Tu \cdot Tv = u \cdot v}$ for all Image may be NSFW.
Clik here to view. ${u \in V_\lambda, v \in V_\mu}$ ), and the extension to generalised eigenvectors then follows from a routine induction. From this we see that Image may be NSFW.
Clik here to view. ${V_1}$ is orthogonal to Image may be NSFW.
Clik here to view. ${V_2}$ , and Image may be NSFW.
Clik here to view. ${V_3}$ and Image may be NSFW.
Clik here to view. ${V_4}$ are null spaces, which by the non-degeneracy of Image may be NSFW.
Clik here to view. {Q} (and hence of the restriction Image may be NSFW.
Clik here to view. ${Q_1}$ of Image may be NSFW.
Clik here to view. {Q} to Image may be NSFW.
Clik here to view. ${V_1}$ ) forces Image may be NSFW.
Clik here to view. ${V_3}$ to have the same dimension as Image may be NSFW.
Clik here to view. ${V_4}$ , indeed Image may be NSFW.
Clik here to view. {Q} now gives an identification of Image may be NSFW.
Clik here to view. ${V_3^*}$ with Image may be NSFW.
Clik here to view. ${V_4}$ . If we let Image may be NSFW.
Clik here to view. ${T_3, T_4}$ be the restrictions of Image may be NSFW.
Clik here to view. {T} to Image may be NSFW.
Clik here to view. ${V_3,V_4}$ , we thus identify Image may be NSFW.
Clik here to view. ${T_4}$ with Image may be NSFW.
Clik here to view. ${T_3^{*-1}}$ , since Image may be NSFW.
Clik here to view. {T} lies in Image may be NSFW.
Clik here to view. {O(Q)} ; in particular Image may be NSFW.
Clik here to view. ${T_3}$ is invertible. Thus

Image may be NSFW.
Clik here to view. $\displaystyle \hbox{det}(1+T_1) = \hbox{det}(1 + T_3) \hbox{det}( 1 + T_3^{*-1} ) = \hbox{det}(T_3)^{-1} \hbox{det}(1+T_3)^2$

and so it suffices to show that Image may be NSFW.
Clik here to view. ${\hbox{det}(T_3) > 0}$ .

At this point we need to use the hypothesis that Image may be NSFW.
Clik here to view. {T} lies in the identity component of Image may be NSFW.
Clik here to view. {O(n,n)} . This implies (by a continuity argument) that the restriction of Image may be NSFW.
Clik here to view. {T} to any maximal-dimensional positive subspace has positive determinant (since such a restriction cannot be singular, as this would mean that Image may be NSFW.
Clik here to view. {T} positive norm vector would map to a non-positive norm vector). Now, as Image may be NSFW.
Clik here to view. ${V_3,V_4}$ have equal dimension, Image may be NSFW.
Clik here to view. ${Q_1}$ has a balanced signature, so Image may be NSFW.
Clik here to view. ${Q_2}$ does also. Since Image may be NSFW.
Clik here to view. ${T_2 = \exp(X_2)}$ , Image may be NSFW.
Clik here to view. ${T_2}$ already lies in the identity component of Image may be NSFW.
Clik here to view. ${O(Q_2)}$ , and so has positive determinant on any maximal-dimensional positive subspace of Image may be NSFW.
Clik here to view. ${V_2}$ . We conclude that Image may be NSFW.
Clik here to view. ${T_1}$ has positive determinant on any maximal-dimensional positive subspace of Image may be NSFW.
Clik here to view. ${V_1}$ .

We choose a complex basis of Image may be NSFW.
Clik here to view. ${V_3}$ , to identify Image may be NSFW.
Clik here to view. ${V_3}$ with Image may be NSFW.
Clik here to view. ${V_3^*}$ , which has already been identified with Image may be NSFW.
Clik here to view. ${V_4}$ . (In coordinates, Image may be NSFW.
Clik here to view. ${V_3,V_4}$ are now both of the form Image may be NSFW.
Clik here to view. ${{\bf C}^m}$ , and Image may be NSFW.
Clik here to view. ${Q( v \oplus w ) = v \cdot w}$ for Image may be NSFW.
Clik here to view. ${v,w \in {\bf C}^m}$ .) Then Image may be NSFW.
Clik here to view. ${\{ v \oplus v: v \in V_3 \}}$ becomes a maximal positive subspace of Image may be NSFW.
Clik here to view. ${V_1}$ , and the restriction of Image may be NSFW.
Clik here to view. ${T_1}$ to this subspace is conjugate to Image may be NSFW.
Clik here to view. ${T_3 + T_3^{*-1}}$ , so that

Image may be NSFW.
Clik here to view. $\displaystyle \hbox{det}( T_3 + T_3^{*-1} ) > 0.$

But since Image may be NSFW.
Clik here to view. ${\hbox{det}( T_3 + T_3^{*-1} ) = \hbox{det}(T_3) \hbox{det}( 1 + T_3^{-1} T_3^{*-1} )}$ and Image may be NSFW.
Clik here to view. ${ 1 + T_3^{-1} T_3^{*-1}}$ is positive definite, so Image may be NSFW.
Clik here to view. ${\hbox{det}(T_3)>0}$ as required. Image may be NSFW.
Clik here to view. $\Box$

The standard branch of the matrix logarithm

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