Faddeev–LeVerrier algorithm: Difference between revisions

Revision as of 23:57, 25 October 2023

Revision as of 23:04, 23 December 2023

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:<math>M_3= A^2-A\mathrm{tr} A -\frac{1}{2}\Bigl(\mathrm{tr} A^2 -(\mathrm{tr} A)^2\Bigr) I,</math>

::<math>c_{n-3}=- \tfrac{1}{6}\Bigl( (\operatorname{tr}A)^3-3\operatorname{tr}(A^2)(\operatorname{tr}A)+2\operatorname{tr}(A^3)\Bigr)=-\frac{1}{3}(c_n \mathrm{tr} A^3+c_{n-1} \mathrm{tr} A^2 +c_{n-2}\mathrm{tr} A); </math>

etc.,<ref>Zadeh, Lotfi A. and Desoer, Charles A. (1963, 2008). ''Linear System Theory: The State Space Approach'' (Mc Graw-Hill; Dover Civil and Mechanical Engineering) {{ISBN|9780486466637}} , pp 303&ndash;305;

</ref><ref>Abdeljaoued, Jounaidi and Lombardi, Henri (2004). ''Méthodes matricielles - Introduction à la complexité algébrique'',

(Mathématiques et Applications, 42) Springer, {{ISBN|3540202471}} .</ref>

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==An equivalent but distinct expression==

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., Fairlie, D. B. and Alshal, H. (2012). "A Galileon Primer", arXiv:1212.6972 , section 3.</ref><ref>{{Cite book|title=Methods of Modern Mathematical Physics|last1=Reed|first1=M.|last2=Simon|first2=B.|publisher=ACADEMIC PRESS, INC.|year=1978|isbn=0-12-585004-2|volume=4 Analysis of Operators|___location=USA|pages=323–333, 340, 343}}</ref>

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