Linear Algebra, 5th Edition

7.4* The Rational Canonical Form

Until now we have used eigenvalues, eigenvectors, and generalized eigenvectors in our analysis of linear operators with characteristic polynomials that split. In general, characteristic polynomials need not split, and indeed, operators need not have eigenvalues! However, the unique factorization theorem for polynomials (see page 562) guarantees that the characteristic polynomial f(t) of any linear operator T on an n-dimensional vector space factors uniquely as

$$f(t)={(-1)}^n{({\varphi }_1(t))}^{n_1}{({\varphi }_2(t))}^{n_2}\cdots {({\varphi }_k(t))}^{n_k},$$

where the ${\varphi }_i(t)$’s $(1\le i\le k)$ are distinct irreducible monic polynomials and the $n_i$’s are positive integers. In the case that f(t) splits, each ...