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In mathematics (linear algebra), the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial of a square matrix, A, named after Dmitry Konstantinovich Faddeev and Urbain Le Verrier. Calculation of this polynomial yields the eigenvalues of A as its roots; as a matrix polynomial in the matrix A itself, it vanishes by the Cayley–Hamilton theorem. Computing the characteristic polynomial directly from the definition of the determinant is computationally cumbersome insofar as it introduces a new symbolic quantity ; by contrast, the Faddeev-Le Verrier algorithm works directly with coefficients of matrix .
The algorithm has been independently rediscovered several times in different forms. It was first published in 1840 by Urbain Le Verrier, subsequently redeveloped by P. Horst, Jean-Marie Souriau, in its present form here by Faddeev and Sominsky, and further by J. S. Frame, and others.[1][2][3][4][5] (For historical points, see Householder.[6] An elegant shortcut to the proof, bypassing Newton polynomials, was introduced by Hou.[7] The bulk of the presentation here follows Gantmacher, p. 88.[8])
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Transcription
The Algorithm
The objective is to calculate the coefficients ck of the characteristic polynomial of the n×n matrix A,
where, evidently, cn = 1 and c0 = (−1)n det A.
The coefficients cn-i are determined by induction on i, using an auxiliary sequence of matrices
It is evidently a matrix polynomial in λ of degree n−1. Thus,
where one may define the harmless M0≡0.
Inserting the explicit polynomial forms into the defining equation for the adjugate, above,
Now, at the highest order, the first term vanishes by M0=0; whereas at the bottom order (constant in λ, from the defining equation of the adjugate, above),
so that shifting the dummy indices of the first term yields
which thus dictates the recursion
for m=1,...,n. Note that ascending index amounts to descending in powers of λ, but the polynomial coefficients c are yet to be determined in terms of the Ms and A.
This can be easiest achieved through the following auxiliary equation (Hou, 1998),
This is but the trace of the defining equation for B by dint of Jacobi's formula,
Inserting the polynomial mode forms in this auxiliary equation yields
so that
and finally
This completes the recursion of the previous section, unfolding in descending powers of λ.
Further note in the algorithm that, more directly,
^Urbain Le Verrier: Sur les variations séculaires des éléments des orbites pour les sept planètes principales, J. de Math. (1) 5, 230 (1840), Online
^Paul Horst: A method of determining the coefficients of a characteristic equation. Ann. Math. Stat.6 83-84 (1935), doi:10.1214/aoms/1177732612
^Jean-Marie Souriau, Une méthode pour la décomposition spectrale et l'inversion des matrices, Comptes Rend.227, 1010-1011 (1948).
^D. K. Faddeev, and I. S. Sominsky, Sbornik zadatch po vyshej algebra (Problems in higher algebra, Mir publishers, 1972), Moscow-Leningrad (1949). Problem 979.
^J. S. Frame: A simple recursion formula for inverting a matrix (abstract), Bull. Am. Math. Soc.55 1045 (1949), doi:10.1090/S0002-9904-1949-09310-2
^Gantmacher, F.R. (1960). The Theory of Matrices. NY: Chelsea Publishing. ISBN0-8218-1376-5.
^Zadeh, Lotfi A. and Desoer, Charles A. (1963, 2008). Linear System Theory: The State Space Approach (Mc Graw-Hill; Dover Civil and Mechanical Engineering) ISBN9780486466637, pp 303–305;
^Abdeljaoued, Jounaidi and Lombardi, Henri (2004). Méthodes matricielles - Introduction à la complexité algébrique,
(Mathématiques et Applications, 42) Springer, ISBN3540202471 .
^Brown, Lowell S. (1994). Quantum Field Theory, Cambridge University Press. ISBN978-0-521-46946-3, p. 54; Also see, Curtright, T. L., Fairlie, D. B. and Alshal, H. (2012). "A Galileon Primer", arXiv:1212.6972, section 3.
^Reed, M.; Simon, B. (1978). Methods of Modern Mathematical Physics. Vol. 4 Analysis of Operators. USA: ACADEMIC PRESS, INC. pp. 323–333, 340, 343. ISBN0-12-585004-2.
Barbaresco F. (2019) Souriau Exponential Map Algorithm for Machine Learning on Matrix Lie Groups. In: Nielsen F., Barbaresco F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science, vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_10
This page was last edited on 23 December 2023, at 23:04