Faddeev-Leverrier Method

Let

be an n × n matrix. The determination of eigenvalues and eigenvectors requires the solution of

(1)

where

is the eigenvalue corresponding to the eigenvector

. The values

must satisfy the equation

(2)

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

(3) [Graphics:Images/FaddeevLeverrierMod_gr_9.gif] .

The Faddeev-Leverrier algorithm is an efficient method for finding the coefficients of the polynomial . As an additional benefit, the inverse matrix is 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) .

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Numerical Analysis

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