Page 216 - Linear Algebra Done Right

Chapter 9. Operators on Real Vector Spaces
206
In the example above, the sum of the multiplicities of the eigenval-
ues of T plus twice the multiplicities of the eigenpairs of T equals 3,
which is the dimension of the domain of T. The next proposition shows
that this always happens on a real vector space.
This proposition shows 9.17 Proposition: If V is a real vector space and T ∈L(V), then
that though an the sum of the multiplicities of all the eigenvalues of T plus the sum
operator on a real of twice the multiplicities of all the eigenpairs of T equals dim V.
vector space may have
no eigenvalues, or it Proof: Suppose V is a real vector space and T ∈L(V). Then there
may have no is a basis of V with respect to which the matrix of T is as in 9.9. The
eigenpairs, it cannot be multiplicity of an eigenvalue λ equals the number of times the 1-by-1
lacking in both these matrix [λ] appears on the diagonal of this matrix (from 9.9). The multi-
2
useful objects. It also plicity of an eigenpair (α, β) equals the number of times x +αx +β is
shows that an operator the characteristic polynomial of a 2-by-2 matrix on the diagonal of this
on a real vector space matrix (from 9.9). Because the diagonal of this matrix has length dim V,
V can have at most the sum of the multiplicities of all the eigenvalues of T plus the sum of
(dim V)/2 distinct twice the multiplicities of all the eigenpairs of T must equal dim V.
eigenpairs.
Suppose V is a real vector space and T ∈L(V). With respect to
some basis of V, T has a block upper-triangular matrix of the form
 
A 1 ∗
 . 
9.18  . .  ,
 
0 A m
where each A j is a 1-by-1 matrix or a 2-by-2 matrix with no eigenval-
ues (see 9.4). We define the characteristic polynomial of T to be the
product of the characteristic polynomials of A 1 ,...,A m . Explicitly, for
each j, define q j ∈P(R) by

x − λ if A j equals [λ];
9.19 q j (x) = ac
(x − a)(x − d) − bc if A j equals bd .
Note that the roots of Then the characteristic polynomial of T is
the characteristic
polynomial of T equal q 1 (x)...q m (x).
the eigenvalues of T,as
Clearly the characteristic polynomial of T has degree dim V. Fur-
was true on complex
thermore, 9.9 insures that the characteristic polynomial of T depends
vector spaces.
only on T and not on the choice of a particular basis.