Weak Existence of Matrix Logarithm

Theorem
Let $T$ be an $n \times n$ matrix and $\|T-I\| < 1$ in the operator norm, with $I$ the  identity.

Then there is a matrix $S$ such that $e^S=T$ where $e^S$ is the matrix exponential.

Proof
Define:
 * $\displaystyle{S = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} (T-I)^n.}$

$\displaystyle{\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} \|T-I\|^n}$ is the Mercator series and converges since $\|T-I\|<1$.

Hence the series for $S$ converges absolutely, and so $S$ is well defined.

Using the series definition for the matrix exponential:

If $c_i=0$ for $i \geq 2$, then $e^S=T$, and the result is shown.

The Mercator series is a Taylor expansion for $\ln(1+x)$ and when combined with the power series of the exponential function gives:

But $e^{\ln(1+x)} = 1+x$.

Thus $1+x = 1 + x + c_2x^2 + c_3x^3 + \ldots$ $\implies$ $c_i=0$ for $i \geq 2$.

Also See

 * Existence of Matrix Logarithm