Analytic function of a matrix: Difference between revisions

Revision as of 07:36, 27 October 2021

Revision as of 15:16, 1 November 2021

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If the [[analytic function]] {{mvar|f}} has the [[Taylor expansion]]

:<math>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 matricesa 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\|\,|B\|</math>.

===Diagonalizable matrices===