Question

求非退化线性替换，把二次型f(x1,x2,x3)=-3x1x2+x1x3-4x2x3化为标准型

Answer 1

解:
f的矩阵a=
1
2
2
2
1
2
2
2
1
|a-λe|
=
(5-λ)(1+λ)^2.
所以a的特征值为
5,
-1,
-1
(a-5e)x
=
0
的基础解系为:
a1
=
(1,
1,
1)'
(a+e)x
=
0
的基础解系为:
a2
=
(1,
-1,
0)',
a3
=
(1,
0,
-1)'
将
a2,a3
正交化得
b2
=
(1,-1,0)',
b3
=
(1/2,1/2,-1)'
单位化得
c1
=
(1/√3,
1/√3,
1/√3)',
c2
=
(1/√2,
-1/√2,
0)',
c3
=
(1/√6,1/√6,-2/√6)'
令矩阵p
=
(c1,c2,c3),
则p为正交矩阵,
且
p^(-1)ap
=
diag(5,-1,-1).
正交变换
x=py,
f
=
5y1^2-y2^2-y3^2.
f不是正定的,
也不负定的.