Faddeev-Leverrier Method

Henceis a root of an nth degree polynomial , which we write in the form

(3).

The Faddeev-Leverrier algorithm is an efficient method for finding the coefficients of the polynomial.As an additional benefit, the inverse matrixis obtained at no extra computational expense.

Recall that the trace of the matrix,written,is

(4).

The algorithm generates a sequence of matrices and uses their traces to compute the coefficients of,

(5)

Then the characteristic polynomial is given by

(6).

In addition, the inverse matrix is given by

(7).

For a proof for equations (6) and (7) see Vera Nikolaevna Faddeeva, "Computational Methods of Linear Algebra," (Translated From The Russian By Curtis D. Benster), Dover Publications Inc. N.Y., Date Published: 1959 ISBN: 0486604241.