Page 218 - Linear Algebra Done Right

Chapter 9. Operators on Real Vector Spaces
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Suppose V is a real vector space and T ∈L(V). Clearly the Cayley-
Hamilton theorem (9.20) implies that the minimal polynomial of T has
degree at most dim V, as was the case on complex vector spaces. If
the degree of the minimal polynomial of T equals dim V, then, as was
also the case on complex vector spaces, the minimal polynomial of T
must equal the characteristic polynomial of T. This follows from the
Cayley-Hamilton theorem (9.20) and 8.34.
Finally, we can now prove a major structure theorem about oper-
ators on real vector spaces. The theorem below should be compared
to 8.23, the corresponding result on complex vector spaces.
9.22 Theorem: Suppose V is a real vector space and T ∈L(V). Let
λ 1 ,...,λ m be the distinct eigenvalues of T, with U 1 ,...,U m the corre-
Either m or M sponding sets of generalized eigenvectors. Let (α 1 ,β 1 ),...,(α M ,β M )
2
might be 0. be the distinct eigenpairs of T and let V j = null(T + α j T + β j I) dim V .
Then

(a) V = U 1 ⊕· · · U m ⊕ V 1 ⊕ ··· ⊕ V M ;
(b) each U j and each V j is invariant under T;
2
(c) each (T − λ j I)| U j and each (T + α j T + β j I)| V j is nilpotent.
This proof uses the Proof: From 8.22, we get (b). Clearly (c) follows from the defini-
same ideas as the proof tions.
of the analogous result To prove (a), recall that dim U j equals the multiplicity of λ j as an
on complex vector eigenvalue of T and dim V j equals twice the multiplicity of (α j ,β j ) as
spaces (8.23). an eigenpair of T. Thus

9.23 dim V = dim U 1 +· · ·+ dim U m + dim V 1 +· · ·+ V M ;

this follows from 9.17. Let U = U 1 +· · ·+ U m + V 1 + ··· + V M . Note
that U is invariant under T. Thus we can define S ∈L(U) by

S = T| U .

Note that S has the same eigenvalues, with the same multiplicities, as T
because all the generalized eigenvectors of T are in U, the domain of S.
Similarly, S has the same eigenpairs, with the same multiplicities, as T.
Thus applying 9.17 to S, we get

dim U = dim U 1 + ··· + dim U m + dim V 1 +· · ·+ V M .