Page 211 - Linear Algebra Done Right

The Characteristic Polynomial
2
null(T + αT + βI) ={0}.
2
(Proof: Because V is one-dimensional, there is a constant λ ∈ R such Recall that α < 4β 201
2
2
that Tv = λv for all v ∈ V. Thus (T + αT + βI)v = (λ + αλ + β)v. implies that
2
2
2
However, the inequality α < 4β implies that λ + αλ + β = 0, and thus x + αx + β has no real
2
null(T + αT + βI) ={0}.) roots; see 4.11.
Now suppose V is a two-dimensional real vector space and T ∈L(V)
has no eigenvalues. If λ ∈ R, then null(T − λI) equals {0} (because T
2
2
has no eigenvalues). If α, β ∈ R with α < 4β, then null(T + αT + βI)
2
equals V if x + αx + β is the characteristic polynomial of the matrix
of T with respect to some (or equivalently, every) basis of V and equals
{0} otherwise (by 9.7). Note that for this operator, there is no middle
2
ground—the null space of T +αT +βI is either {0} or the whole space;
it cannot be one-dimensional.
Now suppose that V is a real vector space of any dimension and
T ∈L(V). We know that V has a basis with respect to which T has
a block upper-triangular matrix with blocks on the diagonal of size at
most 2-by-2 (see 9.4). In general, this matrix is not unique—V may
have many different bases with respect to which T has a block upper-
triangular matrix of this form, and with respect to these different bases
we may get different block upper-triangular matrices.
We encountered a similar situation when dealing with complex vec-
tor spaces and upper-triangular matrices. In that case, though we might
get different upper-triangular matrices with respect to the different
bases, the entries on the diagonal were always the same (though possi-
bly in a different order). Might a similar property hold for real vector
spaces and block upper-triangular matrices? Specifically, is the num-
ber of times a given 2-by-2 matrix appears on the diagonal of a block
upper-triangular matrix of T independent of which basis is chosen?
Unfortunately this question has a negative answer. For example, the
2
operator T ∈L(R ) defined by 9.8 has two different 2-by-2 matrices,
as we saw above.
Though the number of times a particular 2-by-2 matrix might appear
on the diagonal of a block upper-triangular matrix of T can depend on
the choice of basis, if we look at characteristic polynomials instead
of the actual matrices, we find that the number of times a particular
characteristic polynomial appears is independent of the choice of basis.
This is the content of the following theorem, which will be our key tool
in analyzing the structure of an operator on a real vector space.