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this is not always the case over the reals. A polynomial may have no roots at all over
the reals.
1.8.8. Lemma.
Every eigenvalue of a self-adjoint linear transformation T on a
complex vector space
V
with inner product • is real.
Proof.
Let l be an eigenvalue for T and
u
a nonzero eigenvector for l. Then
(
)
=∑=
()
∑=∑
()
=∑
()
=∑ =
(
)
l
uu
∑
l
uu
T
u u u
T
*
u u
T
u u u
l
l
uu
∑
.
Since
u
•
u
π 0, l=
l
, that is, l is real.
1.8.9. Lemma.
Let T be a self-adjoint linear transformation over a real n-
dimensional vector space
V
, n ≥ 1, with inner product •. Then
(1) The characteristic polynomial of T is a product of linear factors.
(2) Eigenvectors corresponding to distinct eigenvalues are orthogonal.
Proof.
By passing to the matrix A for T, (1) follows immediately from Lemma 1.8.8
because we can think of A as defining a complex transformation on
C
n
and every poly-
nomial of degree n factors into linear factors over the complex numbers. To prove (2),
assume that T(
u
) =l
u
and T(
v
) =m
v
for lπm. Then
(
)
=∑=
()
∑= ∑
()
=∑ =
(
)
l
uv
∑
l
uv
T
u v u
T
v u v
m
m
uv
∑
.
Since lπm, it follows that
u
•
v
= 0, and we are done.
1.8.10. Theorem.
(The Real Principal Axes Theorem) Let T be a self-adjoint trans-
formation on an n-dimensional real vector space
V
, n ≥ 1. Then
V
admits an ortho-
normal basis
u
1
,
u
2
,...,
u
n
consisting of eigenvectors of T, that is,
()
=l
T
u
u
i
i
i
for some real numbers l
i.
Proof.
The proof is by induction on n. The theorem is clearly true for n = 1. Assume,
therefore, inductively that it has been proved for dimension n - 1, n > 1. There are
basically two steps involved in the rest of the proof.
First, we need to know that the transformation actually has at least one real eigen-
value l. This was proved by Lemma 1.8.9(1). Let
v
be a nonzero eigenvector for l and
let
u
1
=
v
/|
v
|.
The second step, in order to use the inductive hypothesis, is to show that the
orthogonal complement
W
^
of
W
= <
v
> = <
u
1
> is an invariant subspace of T. This
follows from the fact that if
w
Œ
W
^
, then
()
=∑
()
=
()
∑= ∑=
vw
∑
T
v
T
*
w vwvw
T
l
0
,
