Then the dynamic model of the spindle is established and the

characteristic polynomial is obtained.

0] is a pair of conjugate purely imaginary characteristic roots and substituting [lambda] = i[omega] into the

characteristic polynomial as well as letting f([omega]) = 0, we obtain

A final section describes physical visualizations of eigenvectors--for mirror reflections, rotations, row reductions, and circulant matrices--before turning to the issue of their calculation through the

characteristic polynomial.

The

characteristic polynomial of G is a polynomial of degree n , defined as (G, ) = det (In Eqs.

m](1/t) is the

characteristic polynomial which determines the recurrence relation that [h.

1)--in particular the

characteristic polynomial [gamma]N of A itself--in less than 6N additions (subtractions) and less than 4N multiplications (divisions), it is entirely feasible to find the roots of this polynomial (the eigenvalues of A) by Sturm plus bisection.

Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the

characteristic polynomial which determines the recurrence relation that [h.

Thus, the closed loop

characteristic polynomial is (for simplicity, the operator [q.

S]([lambda]) is the

characteristic polynomial of the shape operator of M, then we have

The Factorization Theorem [9] states that, for any free arrangement A, the

characteristic polynomial of A factors completely over the integers.

2] (say) and the

characteristic polynomial can be written as

i](f) are reflection vectors of the

characteristic polynomial f(z) of the nominal closed-loop system (3).