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If the [[analytic function]] {{mvar|f}} has the [[Taylor expansion]]
<math display="block">f(x) = c_0 + c_1 x + c_2 x^2 + \cdots</math>
then a matrix function <math>A\mapsto f(A)</math> can be defined by substituting {{mvar|x}} by a [[square matrix]]: powers become [[Matrix multiplication#Powers of a matrix|matrix power]]s, additions become matrix sums and multiplications by coefficients become [[scalar multiplication]]s. If the series converges for <math>|x| < r</math>, then the corresponding matrix series converges for matrices {{mvar|A}} such that <math>\|A\| < r</math> for some [[matrix norm]] that satisfies <math>\|AB\|\leq \|A\|\
===Diagonalizable matrices===
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