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  "body": "\n## Definition\n\nLet $f(z) = \\sum_{k=0}^{\\infty} c_k z^k$ be a complex [[power series|0MZB06ST2B]] with radius of convergence $R \u003e 0$, where $R = \\infty$ is allowed.\nLet $n \\ge 1$ and let $A$ be an $n \\times n$ complex [[matrix|0JXZKX526R]] whose [[spectral radius|9V7YQA85PC]] satisfies $\\rho(A) \u003c R$.\nThen the series $\\sum_{k=0}^{\\infty} c_k A^k$ converges, with $A^0 = I$, and its sum is the matrix $f(A)$.\n\n## Remarks\n\nIf $A = P D P^{-1}$ with $D = \\operatorname{diag}(\\lambda_1, \\ldots, \\lambda_n)$, then $f(A) = P \\operatorname{diag}(f(\\lambda_1), \\ldots, f(\\lambda_n)) P^{-1}$.\nThe matrix $f(A)$ is always a polynomial in $A$, so it commutes with $A$.\nExample: $f(z) = 1/(1 - z) = \\sum_k z^k$ has $R = 1$, and $N = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ has $\\rho(N) = 0$.\nSince $N^2 = 0$, $f(N) = I + N$, and indeed $(I - N)(I + N) = I - N^2 = I$.\nNon-example: for the same $f$ and $A = 2I$, $\\rho(A) = 2 \u003e 1$, and the series $\\sum_k 2^k I$ diverges.\nFor $f$ holomorphic near the spectrum of $A$, the Cauchy integral or the Jordan form gives a more general definition; see the holomorphic functional calculus.\n",
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