Faddeev–LeVerrier algorithm: Difference between revisions

Revision as of 18:51, 2 March 2016

Revision as of 16:16, 9 March 2016

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This can be easiest achieved through the following auxiliary equation (Hou, 1998),

::<math>\lambda \frac{\partial p(\lambda) }{ \partial \lambda} -n p =\operatorname{tr} AB~.</math>

This is but the trace of the the defining equation for {{mvar|B}} by dint of [[Jacobi's formula]],

:<math>\frac{\partial p(\lambda)}{\partial \lambda}= p(\lambda) \sum^\infty _{m=0}\lambda ^{-(m+1)} \operatorname{tr}A^m =

p(\lambda) ~ \operatorname{tr} \frac{I}{\lambda I -A}\equiv\operatorname{tr} B~. </math>

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and finally

:<math>\therefore \qquad 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}}.

Further note in the algorithm that, more directly,