While teaching a course on ODE, I needed to introduce the notion of matrix logarithms. I intend to define it as follows.
Definition (Matrix Logarithm)
Let $A\in GL_n(\mathbb{C})$. We define
A) Unipotent case: When $A$ is unipotent, i.e. $A=I+N$, where $N\in M_n(\mathbb{C})$ is nilpotent, we define $\ln A\in M_n(\mathbb{C})$ as $$ \ln A=\ln(I+N):=\sum_{k=1}^{n}\frac{(-1)^{k+1}}{k}N^k. $$ B) Diagonalizable case: When $A$ is diagonalizable, i.e. $A= PDP^{-1}$, where $P\in GL_n(\mathbb{C})$ and $D:=\operatorname*{diag}(\lambda_1,\ldots,\lambda_n)$ with $\lambda_1,\ldots,\lambda_n\in \mathbb{C}\setminus\{0\}$, we define $\ln A\in M_n(\mathbb{C})$ as $$ \ln A:= P\operatorname*{diag}(\ln\lambda_1,\ldots,\ln\lambda_n)P^{-1}, $$ where $\ln \lambda_i = \ln |\lambda_i|+i\arg \lambda_i$, for all $i=1,\ldots,n$.
C) Invertible case: When $A\in GL_n(\mathbb{C})$, we define $$ \ln A:=\ln D + \ln (I+D^{-1}N), $$ where $D, N\in M_n(\mathbb{C})$ are diagonalizable and nilpotent respectively, $A=D+N$, and $DN=ND$. The existence of $D,N$ follows from the Jordan-Chevalley Decomposition Theorem.
The problem arises when I want to prove that $$ e^{\ln A}=A,\text{ for all }A\in GL_n(\mathbb{C}).\ \ \ \ (1) $$
While it is easy to verify equation (1) when $A$ is unipotent or diagonalizable, I don't see any quick way to prove (1) when $A$ is a general invertible matrix. This will be easy to prove if one can show that
$$ \ln D\ln (I+D^{-1}N)=\ln (I+D^{-1}N)\ln D.\ \ \ \ \ (2) $$ Then, the matrix exponential laws and Equation (1) in unipotent or diagonalizable cases will settle (2) in the general case.
QUESTION: Is there any easy way to prove (2), i.e. $$ \ln D\ln (I+D^{-1}N)=\ln (I+D^{-1}N)\ln D, $$ when $D$ is diagonalizable, $N$ is nilpotent and $DN=ND$?
Thank you.
