Page 208 - Linear Algebra Done Right

Chapter 9. Operators on Real Vector Spaces
198
The Characteristic Polynomial
For operators on complex vector spaces, we defined characteristic
polynomials and developed their properties by making use of upper-
triangular matrices. In this section we will carry out a similar procedure
for operators on real vector spaces. Instead of upper-triangular matri-
ces, we will have to use the block upper-triangular matrices furnished
by the last theorem.
In the last chapter, we did not define the characteristic polynomial
of a square matrix with complex entries because our emphasis is on
operators rather than on matrices. However, to understand operators
on real vector spaces, we will need to define characteristic polynomials
of 1-by-1 and 2-by-2 matrices with real entries. Then, using block-upper
triangular matrices with blocks of size at most 2-by-2 on the diagonal,
we will be able to define the characteristic polynomial of an operator
on a real vector space.
To motivate the definition of characteristic polynomials of square
matrices, we would like the following to be true (think about the Cayley-
Hamilton theorem; see 8.20): if T ∈L(V) has matrix A with respect
to some basis of V and q is the characteristic polynomial of A, then
q(T) = 0.
Let’s begin with the trivial case of 1-by-1 matrices. Suppose V is a
real vector space with dimension 1 and T ∈L(V).If [λ] equals the
matrix of T with respect to some basis of V, then T equals λI. Thus
if we let q be the degree 1 polynomial defined by q(x) = x − λ, then
q(T) = 0. Hence we define the characteristic polynomial of [λ] to be
x − λ.
Now let’s look at 2-by-2 matrices with real entries. Suppose V is a
real vector space with dimension 2 and T ∈L(V). Suppose

a c
b d
is the matrix of T with respect to some basis (v 1 ,v 2 ) of V. We seek
a monic polynomial q of degree 2 such that q(T) = 0. If b = 0, then
the matrix above is upper triangular. If in addition we were dealing
with a complex vector space, then we would know that T has charac-
teristic polynomial (z − a)(z − d). Thus a reasonable candidate might
be (x − a)(x − d), where we use x instead of z to emphasize that
now we are working on a real vector space. Let’s see if the polynomial