Calculating the matrix logarithm

The Matrix logarithm of a Linear Operator $X$, is defined as the inverse of the Matrix Exponential, i.e. $\ln (X):=Y$ such that $e^Y=X$. Generally this is not unique, but we can still calculate possible Matrix logarithms for specific cases:

Diagonalisable Matrix

If $X$ is diagonalisable, we can use braket notation to write $X=\sum {\lambda }_i|e_i⟩⟨e_i|$, and then define $\ln (X):=\sum \ln ({\lambda }_i)|e_i⟩⟨e_i|$. Of all eigenvalues ${\lambda }_i$ are positive we can have a preferred solution, however not that if they are note we will obtain complex and non-unique values for $\ln ({\lambda }_i)$.

Mercator series approach

We can use the Mercator series $\ln (1+x)={\sum }_{k=0}^{\infty }\frac{(-1{)}^k}{k}{x}^k$ to obtain a formal expression for the Matrix logarithm. We can obtain an exact logarithm after finitely many sums by:

1. Writing our operator $X$ in Jordan Normal Form
2. Splitting the operator into the individual Jordan blocks. Note that each Jordan block has a unique eigenvalue.
3. For a block $X_k$, calculate $\log (X_k)=\log ({\lambda }_k(I+K))=\log (\lambda I)+\log (I+K)=\log (\lambda )I+{\sum }_{k=0}^{\infty }\frac{(-1{)}^k}{k}{K}^k$. Note that we only need finitely many terms, as $K^m=0\forall m\ge \dim (K)$.