Faddeev–LeVerrier algorithm: Difference between revisions

Revision as of 17:43, 1 March 2016

Revision as of 19:09, 1 March 2016

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</math>

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

&nbsp; <math>\cdots ; \qquad c_{n-m}= -\frac{1}{m}(c_n \mathrm{tr} A^m+c_{n-1} \mathrm{tr} A^{m-1}+...+c_{n-m+1}\mathrm{tr} A); \qquad \cdots </math>

Observe {{math|''A⁻¹ {{=}} − M_n /c₀'' {{=}} (−)^''n''−1''M_n''/det''A''}} terminates the recursion at {{mvar| λ}}. This could be used to obtain the inverse or the determinant of {{mvar|A}}.

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

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!}

\begin{vmatrix} \operatorname{tr}A & m-1 &0&\cdots\\