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第五章 二次型

1．用非退化线性替换化下列二次型为标准形，并利用矩阵验算所得结果。 1）?4x1x2?2x1x3?2x2x3；

2）x2221?2x1x2?2x2?4x2x3?4x3； 3）x221?3x2?2x1x2?2x1x3?6x2x3；

4）8x1x4?2x3x4?2x2x3?8x2x4； 5）x1x2?x1x3?x1x4?x2x3?x2x4?x3x4；

6）x2?2x2212?x4?4x1x2?4x1x3?2x1x4?2x2x3?2x2x4?2x3x4；7）x22221?x2?x3?x4?2x1x2?2x2x3?2x3x4。

解 1）已知 f?x1,x2,x3???4x1x2?2x1x3?2x2x3， 先作非退化线性替换

?x1?y1?y2 ??x2?y1?y2 （1）

??x3?y3则

f?x221,x2,x3???4y1?4y2?4y1y3

??4y22221?4y1y3?y3?y3?4y2

???2y?y32213??y3?4y2， 再作非退化线性替换

??y1?112z1?2z3 ??y2?z2 ??y3?z3?则原二次型的标准形为

f?x2221,x2,x3???z1?4z2?z3，

最后将（2）代入（1），可得非退化线性替换为

2） （

11?x?z?z?z32?1212?11? ?x2?z1?z2?z3 （3）

22??x3?z3??于是相应的替换矩阵为

?11???10??2?110??22????1 T??1?10??010?????001??001??20????????且有

1??2?1?， ?1?201???0??100??? T?AT??040?。

?001???222 2）已知f?x1,x2,x3??x1?2x1x2?2x2?4x2x3?4x3，

由配方法可得

2222 f?x1,x2,x3??x1?2x1x2?x2?x2?4x2x3?4x3

???? ??x1?x2???x2?2x3?，

22于是可令

?y1?x1?x2? ?y2?x2?2x3，

?y?x3?3则原二次型的标准形为

22 f?x1,x2,x3??y1?y2，

且非退化线性替换为

?x1?y1?y2?2y3? ?x2?y2?2y3，

?x?y3?3相应的替换矩阵为

?1?12??? T??01?2?，

?001???

且有

00??110??1?12??100??1???????? T?AT???110??122??01?2???010?。

?2?21??024??00??1????????000?22 （3）已知f?x1,x2,x3??x1?3x2?2x1x2?2x1x3?6x2x3，

由配方法可得

22222 f?x1,x2,x3??x1?2x1x2?2x1x3?2x2x3?x2?x3?4x2?4x2x3?x3

???? ??x1?x2?x3???2x2?x3?，

22于是可令

?y1?x1?x2?x3? ?y2?2x2?x3，

?y?x3?3则原二次型的标准形为

22 f?x1,x2,x3??y1?y2，

且非退化线性替换为

13?x?y?y?y312?122?11? ?x2?y2?y3，

22??x3?y3??相应的替换矩阵为

1??12?1 T??0?2?00??且有

3???2?1??， 2?1???3???2??100??1?????0?10?。 2???1??000??????11T?AT???2?3???21??1??00??1?11??2??110???1?3?3??0???22???1?30??001?1??2??（4）已知f?x1,x2,x3,x4??8x1x2?2x3x4?2x2x3?8x2x4，

先作非退化线性替换

?x1?y1?y4?x?y?22 ?，

x?y3?3??x4?y4则

2 f?x1,x2,x3,x4??8y1y4?8y4?2y3y4?2y2y3?8y2y4

2?2111111????? ?8?y4?2y4?y1?y2?y3???y1?y2?y3??

28??228???2???11??1 ?8?y1?y2?y3??2y2y3

28??2111??1?? ?8?y1?y2?y3?y4??2?y1?y2?y3??2y2y3，

284??2??再作非退化线性替换

222?y1?z1?y?z?z?223 ?，

?y3?z2?z3??y4?z4则

5353??1?? f?x1,x2,x3,x4??8?z1?z2?z3?z4??2?z1?z2?z3?

8844??2??22 ?2z2?2z3，

22再令

53?w?z?x?x312?144??w2?z2 ?，

?w3?z3?153?w4?z1?z2?z3?z4288?则原二次型的标准形为

2222 f?x1,x2,x3,x4???2w1?2w2?2w3?8w4，

且非退化线性替换为

153?x?w?w??121424w3?w4??x2?w2?w3 ?，

?x3?w2?w3?1x??w1?w4?42?相应的替换矩阵为

?1??20 T???0?1???2且有

?54110?3?1?4?10?， ??10?01??0??2??0 T?AT??0??0?00??200?。

0?20??008??（5）已知f?x1,x2,x3,x4??x1x2?x1x3?x1x4?x2x3?x2x4?x3x4， 先作非退化线性替换

?x1?2y1?y2?x?y?22

?，

x?y3?3??x4?y4

则

2 f?x1,x2,x3,x4??2y1y2?y2?2y1y3?2y2y3?2y1y4?2y2y4?y3y4

??y1?y2?y3?y4?再作非退化线性替换

21?32???y3?y4??y4?y12，

2?4?2?z1?y1?z?y?y?y?y21234?? ?， 1?z3?y3?2y4???z4?y4即