Page 183 - Linear Algebra Done Right

Chapter 8. Operators on Complex Vector Spaces
172
8.18
Proposition: If V is a complex vector space and T ∈L(V), then
the sum of the multiplicities of all the eigenvalues of T equals dim V.
Proof: Suppose V is a complex vector space and T ∈L(V). Then
there is a basis of V with respect to which the matrix of T is upper
triangular (by 5.13). The multiplicity of λ equals the number of times λ
appears on the diagonal of this matrix (from 8.10). Because the diagonal
of this matrix has length dim V, the sum of the multiplicities of all the
eigenvalues of T must equal dim V.
Suppose V is a complex vector space and T ∈L(V). Let λ 1 ,...,λ m
denote the distinct eigenvalues of T. Let d j denote the multiplicity
of λ j as an eigenvalue of T. The polynomial

(z − λ 1 ) d 1 ...(z − λ m ) d m

Most texts define the is called the characteristic polynomial of T. Note that the degree of
characteristic the characteristic polynomial of T equals dim V (from 8.18). Obviously
polynomial using the roots of the characteristic polynomial of T equal the eigenvalues
determinants. The of T. As an example, the characteristic polynomial of the operator
2
3
approach taken here, T ∈L(C ) defined by 8.16 equals z (z − 5).
which is considerably Here is another description of the characteristic polynomial of an
simpler, leads to an operator on a complex vector space. Suppose V is a complex vector
easy proof of the space and T ∈L(V). Consider any basis of V with respect to which T
Cayley-Hamilton has an upper-triangular matrix of the form
theorem.
 
λ 1 ∗
 . 
8.19 M(T) =  . .  .
 
0 λ n
Then the characteristic polynomial of T is given by

(z − λ 1 )...(z − λ n );

this follows immediately from 8.10. As an example of this procedure,
3
if T ∈L(C ) is the operator whose matrix is given by 8.17, then the
2
characteristic polynomial of T equals (z − 6) (z − 7).
In the next chapter we will define the characteristic polynomial of
an operator on a real vector space and prove that the next result also
holds for real vector spaces.