Faddeev–LeVerrier algorithm: Difference between revisions

Revision as of 12:56, 27 December 2021

Revision as of 22:07, 26 January 2022

Line 75:

and, in comportance with the [[Cayley–Hamilton theorem]],

:<math> \operatorname{adj}(A) =(-1)^{n-1} M_{n}=(-1)^{n-1} (A^{n-1}+c_{n-1}A^{n-2}+ ...+c_2 A+ c_1 I)=(-1)^{n-1} \sum_{k=1}^n c_k A^{k-1}~.</math>

The final solution might be more conveniently expressed in terms of complete exponential [[Bell polynomials]] as

Line 90 ⟶ 89:

==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=Vol. 4 Analysis of Operators|___location=USA|pages=323-333323–333, 340, 343}}</ref>

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

Line 101 ⟶ 100:

== See also ==

* [[Exterior algebra#Leverrier's%20algorithm algorithm|Exterior algebra § Leverrier's algorithm]]

* [[Jacobi's formula]]

* [[Fredholm determinant]]