Logarithm of a matrix

In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in an element of a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.

The exponential of a matrix A is defined by

Given a matrix B, another matrix A is said to be a matrix logarithm of B if e^A = B.

Because the exponential function is not bijective for complex numbers (e.g. ${\displaystyle e^{\pi i}=e^{3\pi i}=-1}$), numbers can have multiple complex logarithms, and as a consequence of this, some matrices may have more than one logarithm, as explained below. If the matrix logarithm of ${\displaystyle B}$ exists and is unique, then it is written as ${\displaystyle \log B,}$ in which case ${\displaystyle e^{\log B}=B.}$

If B is sufficiently close to the identity matrix, then a logarithm of B may be computed by means of the power series

which can be rewritten as

Specifically, if ${\displaystyle \left\|I-B\right\|<1}$, then the preceding series converges and ${\displaystyle e^{\log(B)}=B}$.

The rotations in the plane give a simple example. A rotation of angle α around the origin is represented by the 2×2-matrix