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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $A$ be an $n \\times n$ complex [[matrix|0JXZKX526R]].\nA *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 $B$.\n\n## Remarks\n\nExample: $A = \\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ has the logarithm $B = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$.\nCheck: $B^2 = 0$, so $e^B = I + B = A$.\nNon-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]].\nA complex matrix has a logarithm if and only if it is [[invertible|0EB9N0S33C]].\nIf $A$ 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 $\\log A$.\nIf $\\norm{A - I} \u003c 1$ for a submultiplicative norm, then $\\log A = \\sum_{k=1}^{\\infty} \\frac{(-1)^{k+1}}{k} (A - I)^k$.\n",
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