3.90
By linearity
𝑓
(
x
) =
∑
𝑗
𝑥
𝑗
𝑓
(
x
𝑗
)
𝑄
defines a quadratic form since
𝑄
(
x
) =
x
𝑇
𝑓
(
x
) =
(
∑
𝑖
𝑥
𝑖
x
𝑖
)
𝑇
⎛
⎝
∑
𝑗
𝑥
𝑗
𝑓
(
x
𝑗
)
⎞
⎠
=
∑
𝑖
∑
𝑗
𝑥
𝑖
𝑥
𝑗
x
𝑇
𝑖
𝑓
(
x
𝑗
) =
∑
𝑖
∑
𝑗
𝑎
𝑖𝑗
𝑥
𝑖
𝑥
𝑗
by Exercise 3.69.
3.91
Let
𝑓
be the symmetric linear operator defining
𝑄
𝑄
(
x
) =
x
𝑇
𝑓
(
x
)
By the spectral theorem (Proposition 3.6), there exists an orthonormal basis
x
1
,
x
2
, . . . ,
x
𝑛
comprising the eigenvectors of
𝑓
. Let
𝜆
1
, 𝜆
2
, . . . , 𝜆
𝑛
be the corresponding eigenvalues,
that is
𝑓
(
x
𝑖
) =
𝜆
𝑖
x
𝑖
𝑖
= 1
,
2
. . ., 𝑛
Then for
x
=
𝑥
1
x
1
+
𝑥
2
x
2
+
⋅ ⋅ ⋅
+
𝑥
𝑛
x
𝑛
𝑄
(
x
) =
x
𝑇
𝑓
(
x
)
= (
𝑥
1
x
1
+
𝑥
2
x
2
+
⋅ ⋅ ⋅
+
𝑥
𝑛
x
𝑛
)
𝑇
𝑓
(
𝑥
1
x
1
+
𝑥
2
x
2
+
⋅ ⋅ ⋅
+
𝑥
𝑛
x
𝑛
)
= (
𝑥
1
x
1
+
𝑥
2
x
2
+
⋅ ⋅ ⋅
+
𝑥
𝑛
x
𝑛
)
𝑇
(
𝑥
1
𝑓
(
x
1
) +
𝑥
2
𝑓
(
x
2
) +
⋅ ⋅ ⋅
+
𝑥
𝑛
𝑓
(
x
𝑛
))
= (
𝑥
1
x
1
+
𝑥
2
x
2
+
⋅ ⋅ ⋅
+
𝑥
𝑛
x
𝑛
)
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Dr.DuMond
 Macroeconomics, Linear Algebra, Negative and nonnegative numbers, Quadratic form, Foundations of Mathematical Economics, orthonormal basis x1, symmetric linear operator

Click to edit the document details