The Matrix logarithm of a Linear Operator , is defined as the inverse of the Matrix Exponential, i.e. such that . Generally this is not unique, but we can still calculate possible Matrix logarithms for specific cases:

Diagonalisable Matrix

If is diagonalisable, we can use braket notation to write , and then define . Of all eigenvalues are positive we can have a preferred solution, however not that if they are note we will obtain complex and non-unique values for .

Mercator series approach

We can use the Mercator series to obtain a formal expression for the Matrix logarithm. We can obtain an exact logarithm after finitely many sums by:

  1. Writing our operator 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 , calculate . Note that we only need finitely many terms, as .