Page 364 - A Course in Linear Algebra with Applications

348 Chapter Nine: Advanced Topics

The minimum polynomial
Let T be a linear operator on an n-dimensional vector
space V over some field of scalars F. We show that T must sat-
isfy some polynomial equation with coefficients in F. At this
point the reader needs to keep in mind the definitions of sum,
scalar multiple and product for linear operators introduced in
n
T
6.3. For any vector v of V, the set {v, (v),..., T (v)} con-
tains n + 1 vectors and so it must be linearly dependent by
5.1.1. Consequently there are scalars ao, i , . . . , a n , not all of
a
them zero, such that

n
a 0 v + aiT(v) + • • • + a nT (v) = 0.
n
/
Let us write v for the polynomial ao+a-\_x + - • • + a nx . Then
n
MT)=a 0l + a 1T+--- + a nT ,
where 1 denotes the identity linear operator. Therefore

n
/vCO(v) = a 0 v + T(v) + ••• + a nT (v) = 0.
ai
Now let {vi,..., v n } be a basis of the vector space V and
define / to be the product of the polynomials / V l , /v 2 > •••) v n •
/
Then
/(r)(vi) = / (r)-../ V n (7)^0 = 0
V l
for each i = l,...,n. This is because f Vi(T)(vi) = 0 and
/
the v (T) commute, since powers of T commute by Exercise
6.3.13. Therefore f(T) is the zero linear transformation on V,
that is,
/CO = o.

Here of course / is a polynomial with coefficients in F.
Having seen that T satisfies a polynomial equation, we
can select a polynomial / in x over F of smallest degree such
that f(T) = 0. In addition, we may suppose that / is monic,