where is the eigenvalue corresponding to the eigenvector . The values must satisfy the equation
Hence is a root of an nth degree polynomial , which we write in the form
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
The algorithm generates a sequence of matrices and uses their traces to compute the coefficients of ,
Then the characteristic polynomial is given by
In addition, the inverse matrix is given by