---
tag: 90MF8M915N
kind: definition
name: Logarithm of a matrix
msc: [15A16]
defines: FS339MCFEQ
uses: [0JXZKX526R, 020M04FN1H]
sources:
  - url: https://en.wikipedia.org/wiki/Logarithm_of_a_matrix
---

## Definition

Let $n \ge 1$ and let $A$ be an $n \times n$ complex [[matrix|0JXZKX526R]].
A *logarithm* of $A$ is an $n \times n$ complex matrix $B$ such that $e^B = A$, where $e^B$ is the [[matrix exponential|020M04FN1H]] of $B$.

## Remarks

Example: $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ has the logarithm $B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$.
Check: $B^2 = 0$, so $e^B = I + B = A$.
Non-example: the zero matrix has no logarithm, because $\det e^B = e^{\tr B} \ne 0$ for every $B$, by the [[determinant of a matrix exponential|0A1QQN333A]].
A complex matrix has a logarithm if and only if it is [[invertible|0EB9N0S33C]].
If $A$ has no eigenvalue in $(-\infty, 0]$, there is a unique logarithm whose eigenvalues have imaginary part in $(-\pi, \pi)$; it is the *principal logarithm*, written $\log A$.
If $\norm{A - I} < 1$ for a submultiplicative norm, then $\log A = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} (A - I)^k$.