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In-Depth Information
The solutions to these equations have the form x(1,1,1). Let
u
3
= (1/÷3,1/÷3,1/÷3).
Finally, if P is the matrix whose columns are the
u
i
, then
1
2
1
6
1
3
Ê
ˆ
-
-
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
100
010
004
Ê
ˆ
1
2
1
6
1
3
Á
Á
˜
˜
-
1
P
=
-
and
P
AP
=
Ë
¯
2
6
1
3
0
Ë
¯
Definition.
A linear transformation T is said to be
normal
if it commutes with its
adjoint, that is, TT* = T*T.
1.8.14. Theorem.
(The Complex Principal Axes Theorem) Let T be a normal trans-
formation on an n-dimensional complex vector space
V
, n ≥ 1. Then
V
admits an
orthonormal basis
u
1
,
u
2
,...,
u
n
consisting of eigenvectors of T, that is,
()
=l
T
u
u
i
i
i
for some complex numbers l
i.
Proof.
See [Lips68].
The matrix form of Theorem 1.8.14 is
1.8.15. Theorem.
If A is a normal matrix, then there exists an unitary matrix P so
that D = P
-1
AP is a diagonal matrix.
Proof.
See [Lips68].
1.9
Bilinear and Quadratic Maps
This section describes some maps that appear quite often in mathematics. However,
we are not interested in just the general theory. Quadratic maps and quadratic forms,
in particular, have important applications in a number of areas of geometry and topol-
ogy. For example, the conics, which are an important class of spaces in geometry, are
intimately connected with quadratic forms. Other applications are found in Chapters
8 and 9.
Definition.
A
bilinear map
on a vector space
V
over a field k is a function
f:
V
¥
V
Æ k satisfying
(1) f(a
v
+ b
v
¢,
w
) = af(
v
,
w
) + bf(
v
¢,
w
), and
(2) f(
v
,a
w
+ b
w
¢) = af(
v
,
w
) + bf(
v
,
w
¢),
for all
v
,
v
¢,
w
,
w
¢Œ
V
and a, b Πk.
