Faddeev–LeVerrier algorithm: Difference between revisions

Revision as of 16:16, 9 March 2016

Revision as of 13:34, 10 March 2016

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

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

:<math> c_{n-k} = \frac{(-1)^{n-k}}{k!} B_k \Bigl ( \operatorname{tr}A , -1! ~ \operatorname{tr}A^2, 2! ~\operatorname{tr}A^3, \ldots, (-1)^{k-1}(k-1)! ~ \operatorname{tr}A^k\Bigr ) .</math>

==An equivalent but distinct expression==