Analytic function of a matrix: Difference between revisions

Revision as of 15:56, 22 December 2018

Revision as of 07:45, 14 January 2019

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The above Taylor power series allows the scalar <math>x</math> to be replaced by the matrix. This is not true in general when expanding in terms of <math>A(\eta) = A+\eta B</math> about <math>\eta = 0</math> unless <math>[A,B]=0</math>. A counter example is <math>f(x) = x^{3}</math>, which has a finite length Taylor series. We compute this in two ways,

* Brute force :

:<math>f(A+\eta B) = (A+\eta B)^{3} = A^{3} + \eta(A^{2}B + ABA + BA^{2}) + \eta^{2}(AB^{2} + BAB + B^{2}A) + \eta^{3}B^{3}</math>