1)4x1x2 2x1x32x2 x3;2)2 c C 2x1 2x1x2 2x24x2x3 4x2;3)x； 3x2 2x1 x22x1 x3 6x2x3 ；4)8x1x4 2x3 x4 2x2x3 8x2x4 ；5)x1x2 x1x3 x1x4 x2x3 x2x4 x3x4;6)2c 22, " C C C Cx1 2x2 x4 4x1x2 4x1x3 2x1x4 2x2x3 2x2M 2x3x4 ;7)x； x； x2 x2 2x1x2 2x2x3 2x3x4解1）已知f ”,乂2?34x1x22x1x3 2x2x3 ,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最后将（2）代入（1）,可得非退化线性替换为先作非退化线性替换x1y1y2x2y1y2(1)x3y34y124y2 4y1y34y124y1y32Y34y；2y13y3再作非退化线性替换y1y212 z1z212 z3(2)y3Z3则原二次型的标准形为x1,x2,x3Z24z211XiZlZ2Z322(3)1 1X2-Z1Z2-Z32 2X3Z31 1 2T 0 1 20 0 1于是相应的替换矩阵为1212012121且有100TAT040。
0012 )已知 f X1 ,X2,X3xj 2x1x2 2X2 4x2x3 4X2,由配方法可得f X1, X2, X32X12x1x22X22 2x2 4x2x3 4x3X1X2X2于是可令y1X1y2X2X22x3 ,y3X32V2则原二次型的标准形为X1,X2,X3且非退化线性替换为X1y1y22y3X2y22y3X3y3相应的替换矩阵为0 且有TAT(3)已知X1,X2,X32X13x22x1x22x1X36x2x3,由配方法可得fX1,X2,X32X12x1x22X1X32x2X32X22X34x24X2X3x32于是可令X1X2X322x2X3则原二次型的标准形为且非退化线性替换为X1X2X3相应的替换矩阵为且有TATy1y2y3y1y3X12x2X3y2X2X3X3X1,X2,X32Y12V23y一、3212y32 2(4)已知 f x1,x2,x3,x4 8x1x2 2x3x4 2x2x3 8x2x4 ,先作非退化线性替换xiyiy4X2y2X3y3X4、4X1,X2.X3.X428y1y48y42y3y42y2y38y2y42y42y4yii2y2i8y3i2yii2y2再作非退化线性替换fXi,X2,X3,X4再令则原二次型的标准形为y3i2yii2yi8Tzi2Z2w2W3W4i2y2i2y2yiy2y3y4Z2ZiZ2Z2Z45Z282Z2Z3i2Zi5X24y3i8y3Z3Z33Z3858Z2Xi,X2,X3,X42y2y3V4Z4yiZiV25Z243X3438Z3i4y33Z342y2y3,2w；2w；2w；8w2,Z4且非退化线性替换为W4相应的替换矩阵为且有X3X42T001TAT153-W1W2W3244W2W3W2W3XiX21W1W4(5)已知fx1,x2,x3,x4先作非退化线性替换fx1，x2,x3,x4再作非退化线性替换x〔x2x/3x〔x4x?x3x?x4x3x4x2x3x42y1y2y2y3y，22y』2V22y』32y2y32丫区2y2y，2%V2V3y4y1Z2Z3y〔yy31V33y4V4Z4V4则原二次型的标准形为V2y3V4fX1,X2,X3,X4且非退化线性替换为XiX2X3X4相应的替换矩阵为且有(6)已知由配方法可得TATX1，X2,X3,X4fX1,X2,X3,X42X11Z3-Z42Z42Z12Z2Z2ZiZ2Z3Z42x1Z32Z3Z3Z312Z41—Z4212Z4，1一Z4212121212x；2X44x1x24印32x1x42X2X32x2x42X3X42x12x22x3x42x22X32X42x2 2x3 x4 2 2x2 X2 2X2X3 2x2x4 2X3X42于是可令则原二次型的标准形为且非退化线性替换为故替换矩阵为且有(7)已知f由配方法可得Xi2x222x3x42x2y1x12x22x3x4y2x2V4x3x431—x3x42212X331-x3-x422x4x1V12y2V33x2V22y3V4x3V3V4x4V42c212V12y2-y3,2V4TAT13210010200为“2?3,乂422x〔x22x32M2x1x22x2x32x3x4,f2,乂2?3442x22x2x1x3x1x322x1x32x3x4x22x2x32x1x3222x32x3x4x4x3于是可令x1x2x3x3x42x1x3x；x2x；x1x22x3x3则原二次型的标准形为且非退化线性替换为相应的替换矩阵为y2X1X2X3y3X3X4y，X1X32222y1y2y2y4X1y1X2y2y，X3y1y，X4y1y3y，10000101yi%且有(1) 在实数域上，若作非退化线性替换TAT(n)把上述二次型进 退化线性替换。
步化为规范形,分实系数、复系数两种情形；并写出所作的非解1)已求得二次型fX1,X2,X34X1X22X1X32X2X3的标准形为且非退化线性替换为X1X2X32.22fy14y23y3,11二y1y2二y32211二，y2y22y3yiV2Z312Z2’V3可得二次型的规范形为222ZiZ2Z32) 在复数域上,若作非退化线性替换yiV2iZi 1 2Z2’y3Zi可得二次型的规范形为2Zi2 2Z2 Z32)已求得二次型f X1,X2,X32Xi2 22x1x2 2x2 4x2x3 4x3的标准形为且非退化线性替换为XiyiX2y2y2 2 y32 y3 ,X3y3故该非退化线性替换已将原二次型化为实数域上的规范形和复数域上的规范形fy；y23)已求得二次型Xi, X2, X32Xi3x22x1x22X1X36x2x3XiX2yi的标准形为22fyiy2,且非退化线性替换为3y2y32(1) 在实数域上，上面所作非退化线性替换已将二次型化为规范形，即22fyiy22) 在复数域上，若作非退化线性替换yiziy2iz2yZ3可得二次型的规范形为fZi2Z23)已求得二次型fXi,X2,X3,X48X1X22X3X42X2X38X2X4的标准形为_2_2_2_2f2y；2y22y28y2,且非退化线性替换为i5Xi-yi-V2X2V2y3V4X3V2V3iX4-yiy4(i)在实数域上，若作非退化线性替换yiy2y3y4可得二次型的规范形为iZ4.2i一Z2,2i2z3i2、2乙2222ZiZ2Z3Z2。
2)在复数域上，若作非退化线性替换yiV2V3。