Question

如何用初等变换化二次型q(x1,x2,x3)=x1^2+2x1x2+2x2^2+4x2x3+4x3^2为标准型,并写出相应的线性替换？

Answer 1

解: 二次型的矩阵 A =
1 1 0
1 2 2
0 2 4

分块矩阵 (上下放置)
A
E
=
1 1 0
1 2 2
0 2 4
1 0 0
0 1 0
0 0 1

r2-r1,c2-c1
1 0 0
0 1 2
0 2 4
1 -1 0
0 1 0
0 0 1

r3-2r2,c3-2c2
1 0 0
0 1 0
0 0 0
1 -1 2
0 1 -2
0 0 1

令 P=
1 -1 2
0 1 -2
0 0 1

X=PY, f = y1^2+y2^2.

验证: P^TAP = diag(1,1,0).