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Kunze Linear Algebra
· Problem 10.2.6
Bilinear Forms
Mathematics — Methods · diagonalizing-quadratic-form
Problem
Let
q
q
q
be the quadratic form on
R
2
R^2
R
2
given by
q
(
x
1
,
x
2
)
=
2
b
x
1
x
2
.
q(x_1, x_2) = 2bx_1 x_2.
q
(
x
1
,
x
2
)
=
2
b
x
1
x
2
.
Find an invertible linear operator
U
U
U
on
R
2
R^2
R
2
such that
(
U
†
q
)
(
x
1
,
x
2
)
=
2
b
x
1
2
−
2
b
x
2
2
.
(U^\dagger q)(x_1, x_2) = 2bx_1^2 - 2bx_2^2.
(
U
†
q
)
(
x
1
,
x
2
)
=
2
b
x
1
2
−
2
b
x
2
2
.
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