The Exact Computation of the Characteristic Polynomial of a Matrix

It is not recommended, in general, that we compute the coefficients of the characteristic polynomial of a matrix as a first step in finding the eigenvalues of the matrix by polynomial root-finding techniques. This is due to the fact that if ordinary floating-point arithmetic is used, the accumulation of rounding errors will produce only approximations to the coefficients and, if the polynomial is ill-conditioned, the roots of the “approximate characteristic equation” may not be good approximations to the roots of the characteristic equation. See Wilkinson [1963], Chapter 2, for a discussion of the condition of a polynomial equation with respect to the computation of its roots.