Page 365 - A Course in Linear Algebra with Applications

9.4: Jordan Normal Form 349

that is, the highest power of x in / has its coefficient equal to
1. This polynomial / is called a minimum polynomial of T.
Suppose next that g is an arbitrary polynomial with coef-
ficients in F. Using long division, just as in elementary algebra,
we can divide g by / to obtain a quotient q and a remainder r;
both of these will be polynomials in x over F. Thus g = fq+r,
and either r = 0 or the degree of r is less than that of . Then
/
we have
g(T) = f(T)q(T) + r(T) = r(T)

since f(T) = 0. Therefore g(T) = 0 if and only if r(T) = 0.
But, remembering that / was chosen to be of smallest degree
subject to f(T) = 0, we can conclude that r(T) = 0 if and
/
only if r = 0, that is, g is divisible by . Thus the polynomials
that vanish at T are precisely those that are divisible by the
polynomial /.
If g is another monic polynomial of the same degree as
/
/ such that g(T) = 0, then in fact g must equal . For g is
divisible by / and has the same degree as , which can only
/
/
mean that g is a constant multiple of . However g is monic,
so it actually equals . Therefore the minimum polynomial of
/
T is the unique monic polynomial / of smallest degree such
that f(T) = 0.
These conclusions are summed up in the following result.

Theorem 9.4.1
Let T be a linear operator on a finite-dimensional vector space
over a field F with a minimum polynomial f. Then the only
polynomials g with coefficients in F such that g(T) = 0 are
the multiples of f. Hence f is the unique monic polynomial
of smallest degree such that f(T) = 0 and T has a unique
minimum polynomial.

So far we have introduced the minimum polynomial of
a linear operator, but it is to be expected that there will be