高等代数 第四讲

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1、pê:1oÙg.QuadraticForms1pê1oÙg.QuadraticForms!,QingjieYang2011Version2.1–2011!,QingjieYangpê:1oÙg.QuadraticForms21–Introduction.........................................82–StandardForms.......................................223–Uniqueness...............

2、..........................574–PositiveDefinedQuadraticForms..............................73Version2.1–2011!,QingjieYangpê:1oÙg.QuadraticForms3¯KJÑ)ÛAÛ¥,²¡þg§22ax+2bxy+cy+2dx+2ey+f=0.Version2.1–2011!,QingjieYangpê:1oÙg.QuadraticForms4☞b=0,^{z22′a(x−x0

3、)+c(y−y0)=f,☞b6=0,I¶^=O′′x=xcosθ−ysinθ,y=x′sinθ+y′cosθ.,zþ¡/′′2′′2′′′′ax+cy+2dx+2ey+f=0.Version2.1–2011!,QingjieYangpê:1oÙg.QuadraticForms53êÆ©Û¥,4¯K:k¼êf(x,y)÷v′′fx(a,b)=fy(a,b)=0ëY.@of(x,y)3(a,b)?Taylorúªf(x,y)=f(a,b)∂2f(x−a)2∂2f∂2f(y−b)2+

4、+(x−a)(y−b)++R∂x22!∂x∂y∂y22!(a,b)(a,b)(a,b)Version2.1–2011!,QingjieYangpê:1oÙg.QuadraticForms6Pu=x−a,v=y−b,1∂2f1∂2f1∂2fa11=,a12=,a22=;2∂x22∂x∂y2∂y2(a,b)(a,b)(a,b)K4¯K=zg(u,v)3(0,0)?4¯K:22g(u,v)=a11u+2a12uv+a22v.➳a12=0,éN´ä;➳a126=0,Ia12.Version2.1–20

5、11!,QingjieYangpê:1oÙg.QuadraticForms7ÙÌSN:☞g.ÄVg☞g.ÝL«☞òz5O,{☞ÝÜÓ☞g.IO/☞g.5/☞½g.Ú½ÝVersion2.1–2011!,QingjieYangpê:1oÙg.QuadraticForms81–g.ÝL«(Introduction)§1IntroductionVersion2.1–2011!,QingjieYangpê:1oÙg.QuadraticForms9½Â1.êPþ'ux1

6、,x2,...,xngàgõªf(x1,x2,...,xn)2=a11x1+2a12x1x2+2a13x1x3+···+2a1nx1xn2+a22x2+2a23x2x3+···+2a2nx2xn···························2+annxn.¡Pþng.(quadraticform),{¡g..§1IntroductionVersion2.1–2011!,QingjieYangpê:1oÙg.QuadraticForms10~1.4x2−6xx+4xx+2x2−x2´n

7、g..1121323§1IntroductionVersion2.1–2011!,QingjieYangpê:1oÙg.QuadraticForms11x1,x2,...,xny1,y2,...,yn5O:x1=c11y1+c12y2+···+c1nynx2=c21y1+c22y2+···+c2nyn(1)............xn=cn1y1+cn2y2+···+cnnyn§1IntroductionVersion2.1–2011!,QingjieYangpê:1

8、oÙg.QuadraticForms12·K1.5Org.zg..5P.·ØU^?¿5Oz{g.,ÄKCzg.¬Nõ5,~X:xi=0,i=1,...,n,Kf(x1,x2,...,xn)=0.§1IntroductionVersion2.1–2011!,QingjieYangpê:1oÙg.QuadraticForms13½Â2.XJXê1ª