Logarithm of a matrix

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. A complex matrix has a logarithm if and only if it is invertible. 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 $\log A = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} (A - I)^k$.