Faddeev–LeVerrier algorithm


In mathematics, the Faddeev–LeVerrier algorithm is a recursive method to calculate the coefficients of the characteristic polynomial of a square matrix,, named after Dmitry Konstantinovich Faddeev and Urbain Le Verrier. Calculation of this polynomial yields the eigenvalues of as its roots; as a matrix polynomial in the matrix itself, it vanishes by the fundamental Cayley–Hamilton theorem. Calculating determinants, however, is computationally cumbersome, whereas this efficient algorithm is computationally significantly more efficient.
The algorithm has been independently rediscovered several times, in some form or another. 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.

The Algorithm

The objective is to calculate the coefficients of the characteristic polynomial of the matrix,
where, evidently, = 1 and 0 = n det.
The coefficients are determined recursively from the top down, by dint of the auxiliary matrices,
Thus,
etc.,
...;
Observe terminates the recursion at. This could be used to obtain the inverse or the determinant of.

Derivation

The proof relies on the modes of the adjugate matrix,, the auxiliary matrices encountered.
This matrix is defined by
and is thus proportional to the resolvent
It is evidently a matrix polynomial in of degree. Thus,
where one may define the harmless ≡0.
Inserting the explicit polynomial forms into the defining equation for the adjugate, above,
Now, at the highest order, the first term vanishes by =0; whereas at the bottom order,
so that shifting the dummy indices of the first term yields
which thus dictates the recursion
for =1,...,. Note that ascending index amounts to descending in powers of, but the polynomial coefficients are yet to be determined in terms of the s and.
This can be easiest achieved through the following auxiliary equation,
This is but the trace of the defining equation for 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.
Further note in the algorithm that, more directly,
and, in comportance with the Cayley–Hamilton theorem,
The final solution might be more conveniently expressed in terms of complete exponential Bell polynomials as

Example

Furthermore, , which confirms the above calculations.
The characteristic polynomial of matrix is thus ; the determinant of is ; the trace is 10=−c2; and the inverse of is

An equivalent but distinct expression

A compact determinant of an ×-matrix solution for the above Jacobi's formula may alternatively determine the coefficients,