Analytic function of a matrix: Difference between revisions

Revision as of 23:30, 18 October 2021

Revision as of 07:36, 27 October 2021

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=== Cauchy integral ===

[[Cauchy's integral formula]] from [[complex analysis]] can also be used to generalize scalar functions to matrix functions. Cauchy's integral formula states that for any [[analytic function]] {{mvar|f}} defined on a set {{math|''D'' ⊂ ℂ<math>\mathbb{C}</math>}}, one has

:<math>f(x) = \frac{1}{2\pi i} \oint_{C}\! {\frac{f(z)}{z-x}}\, \mathrm{d}z ~,</math>

where {{mvar|C}} is a closed simple curve inside the ___domain {{mvar|D}} enclosing {{mvar|x}}.

Now, replace {{mvar|x}} by a matrix {{mvar|A}} and consider a path {{mvar|C}} inside {{mvar|D}} that encloses all [[eigenvalue]]s of {{mvar|A}}. One possibility to achieve this is to let {{mvar|C}} be a circle around the [[origin (mathematics)|origin]] with [[radius]] larger than ‖{{mvar|A}}‖ for an arbitrary [[matrix norm]] ‖•‖. Then, {{math|''f''&hairsp;{{hair space}}(''A'')}} is definable by

:<math>f(A) = \frac{1}{2\pi i} \oint_C f(z)\left(zI-A\right)^{-1} \mathrm{d}z~. </math>

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The convergence criteria of the power series then apply, requiring <math>\Vert \eta A^{-1}B \Vert</math> to be sufficiently small under the appropriate matrix norm. For more general problems, which cannot be rewritten in such a way that the two matrices commute, the ordering of matrix products produced by repeated application of the Leibniz rule must be tracked.

=== Arbitrary function of a 2&times;22×2 matrix ===

An arbitrary function ''f(A)'' of a 2×2 matrix A has its [[Sylvester's formula]] simplify to

:<math>