Page 212 - Linear Algebra Done Right

Chapter 9. Operators on Real Vector Spaces
202
9.9
Theorem:
Suppose V is a real vector space and T ∈L(V).
Suppose that with respect to some basis of V, the matrix of T is
 
A 1 ∗
 . 
9.10  . .  ,
 
0 A m
where each A j is a 1-by-1 matrix or a 2-by-2 matrix with no eigenvalues.
(a) If λ ∈ R, then precisely dim null(T − λI) dim V of the matrices
A 1 ,...,A m equal the 1-by-1 matrix [λ].
2
This result implies that (b) If α, β ∈ R satisfy α < 4β, then precisely
2
null(T + αT + βI) dim V dim null(T + αT + βI) dim V
2
must have even
2
dimension.
of the matrices A 1 ,...,A m have characteristic polynomial equal
2
to x + αx + β.
This proof uses the Proof: We will construct one proof that can be used to prove both
2
same ideas as the proof (a) and (b). To do this, let λ, α, β ∈ R with α < 4β. Define p ∈P(R)
of the analogous result by

on complex vector x − λ if we are trying to prove (a);
p(x) = 2
spaces (8.10). As usual, x + αx + β if we are trying to prove (b).
the real case is slightly Let d denote the degree of p. Thus d = 1 if we are trying to prove (a)
more complicated but and d = 2 if we are trying to prove (b).
requires no new We will prove this theorem by induction on m, the number of blocks
creativity. along the diagonal of 9.10. If m = 1, then dim V = 1 or dim V = 2; the
discussion preceding this theorem then implies that the desired result
holds. Thus we can assume that m> 1 and that the desired result
holds when m is replaced with m − 1.
For convenience let n = dim V. Consider a basis of V with respect
to which T has the block upper-triangular matrix 9.10. Let U j denote
the span of the basis vectors corresponding to A j . Thus dim U j = 1
if A j is a 1-by-1 matrix and dim U j = 2if A j is a 2-by-2 matrix. Let
U = U 1 + ··· + U m−1 . Clearly U is invariant under T and the matrix
of T| U with respect to the obvious basis (obtained from the basis vec-
tors corresponding to A 1 ,...,A m−1 )is
 
A 1 ∗
 . 
9.11  . .  .
 
0 A m−1