Faddeev–LeVerrier algorithm: Difference between revisions

Revision as of 19:09, 1 March 2016

Revision as of 20:52, 1 March 2016

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and finally

:<math> c_{n-m} = -\frac{1}{m} \operatorname{tr}A M_{m} ~.</math>

This completes the recursion of the previous section, unfolding in descending powers of {{mvar|λ}}.

This completes the recursion of the previous section, unfolding in descending powers of {{mvar|λ}}. Further note in the algorithm that, more directly,

:<math> M_{m} =A M_{m-1} - \frac{1}{m-1} (\operatorname{tr}A M_{m-1}) I ~.,</math>

and

:<math> B =(-)^{n-1} M_{n}=(-)^{n-1} (A^{n-1}+c_{n-1}A^{n-2}+ ...+c_2 A+ c_1 I)~.</math>

<br>

A compact determinant of an {{mvar|m}}×{{mvar|m}}-matrix solution for the above Jacobi's formula may alternatively determine the coefficients {{mvar|c}}<ref>Brown, Lowell S. (1994). ''Quantum Field Theory'', Cambridge University Press. ISBN 978-0-521-46946-3, p. 54; Also see, Curtright, T. L. and Fairlie, D. B. (2012). "A Galileon Primer", arXiv:1212.6972 , section 3.</ref>

:<math>c_{n-m} = \frac{(-1)^m}{m!}