用合同变换法化下列二次型为标准型,并写出所用的替换矩阵f(x1,x2,x3)=x2^2+x3^2+4x1x2+2x1x3+2x2x3

题目

用合同变换法化下列二次型为标准型,并写出所用的替换矩阵f(x1,x2,x3)=x2^2+x3^2+4x1x2+2x1x3+2x2x3

答案

(A;E)=
0 2 1
2 1 1
1 1 1
1 0 0
0 1 0
0 0 1
r1-r3,r2-r3
-1 1 0
1 0 0
0 0 1
1 0 0
0 1 0
-1 -1 1
r2+r1
-1 0 0
0 1 0
0 0 1
1 1 0
0 1 0
-1 -2 1
C=
1 1 0
0 1 0
-1 -2 1
X=CY,
f=-y1^2 + y2^2 + y3^2