1、?g.4.5?g.9?LHO?g.?g.?g.?P?u n C x1,xn?gga11x21+2a12x1x2+2a12x1x3+2a1nx1xn+a22x22+2a23x2x3+2a2nx2xn+annx2n P?n?g.,?g.Pf(x1,xn)f.?g.?g.?g.?d?g.?,?i j,aijk.e?g.f,?i j,P aij=aji.uf(x1,xn)=a11x21+a12x1x2+a1nx1xn+a21x2x1+a22x22+a2nx2xn+an1xnx1+an2xnx2+annx2n=nXi,j=1aijxixj.,|?B/L?g.f(x1,xn)=x1(a11x1+a12x2
2、+a1nxn)+x2(a21x1+a22x2+a2nxn)+xn(an1x1+an2x2+annxn)?g.?g.?g.?=(x1,x2,xn)a11x1+a12x2+a1nxna21x1+a22x2+a2nxnan1x1+an2x2+annxn=(x1,x2,xn)a11a12a1na21a22a2n.an1an2annx1x2.xn=XTAX.u?g./?f=XTAX,A?g.f?.?g.?g.dc?5 aij=aji,i,j=1,n,AT=A.w,?g.?.?g.,?(?;?,(?g.?g.?A.?A?g.f?,P r(f).?g.?g.f?g.f(x1,x2,x3)=x21 4x1x2
3、 6x1x3+2x22 x2x3?.)a11=1,a21=a12=2,a31=a13=3,a22=2,a32=a23=12,a33=0.?g.f?12322123120.?g.?g.f?A=0112120010332030.?g.)?A L?g.f(x1,x2,x3,x4)=(x1,x2,x3,x4)0112120010332030 x1x2x3x4=2x1x2+2x1x3+4x1x4+2x22+3x23+6x3x4?g.?g.P?B,ePY=y1y2.ynZ=z1z2.znW=w1w2.wn.?g.?g.5O?x1,x2,xn;y1,y2,yn|C,Xx1=c11y1+c12y2+c1nyn
4、,x2=c21y1+c22y2+c2nyn,xn=cn1y1+cn2y2+cnnyn(1.1)d x1,x2,xn?y1,y2,yn?5O,5O.eX1?|cij|6=0,K5O(1.1)zzz?.?g.?g.?L5O3x1=c11y1+c12y2+c1nyn,x2=c21y1+c22y2+c2nyn,xn=cn1y1+cn2y2+cnnyn-C=1cn2cnn,X=x1x2.xn,Y=y1y2.ynK5OL X=CY.e C _,K_5O,e C?,K?5O.?g.?g.?g.5O?f(x1,x2,xn)=XTAX?g.z5OX=CY,?u y1,y2,yn?g.YTBY.5L5O X=CY
5、?,A B m?X.f(x1,x2,xn)=XTAX=(CY)TA(CY)=YT(CTAC)Y=g(y1,y2,yn)=YTBY.CTAC?,u B=CTAC.?g.?g.?X?A,B Pnn,e3 P?_?C?B=CTAC,K A B?.?dddXXX.ggg555 A=ETAE;555 dB=CTAC=A=(C1)TBC1;DDD444555 dA1=CT1AC1 A2=CT2A1C2=A2=(C1C2)TA(C1C2).?g.?g.o(Lz?5O#?g.?g.?.Ur?g.z?g.?duT?g.?u?.?g.?,kf(X)=f(CY)=d1y21+d2y22+dny2n./?g.?IO/
6、.eIO/?d1,dn1,0,K?g.?5/.?g.?g.?5?k?,=A B,K r(A)=r(B).L_5O X=CY?,?g.f(X)=XTAX?d A C CTAC,r(f)C.e A,A B,K B?.?g.?g.?n P?.yyy?A=(aij)nn?.n 8B.(1)?n=1,(w,.(2)b?(u n 1?.ey(u n?.”?A 6=0 a116=0.PA=a11TA1!,C1=11a110En1!KCT1AC1=101a11TEn1!a11TA1!11a110En1!?g.?g.?=a1100A2!A2=A11a11TEn1.N?y A2?,|8Bb?,3_?C2?CT2A
7、2C2?.-C=C1 100C2!K C _?,CTAC=100CT2!CT1AC1 100C2!=100CT2!a1100A2!100C2!=a1100CT2A2C2!d CTAC?.?g.4.6?g.?IO/HO?g.?P?g.?!0?P?g.?IO/9E C R?g.?IO/.kw?P?g.,d!?(=?nef(X)=XTAX=nXi,j=1aijxixj P?g.,K3_5O X=CY,f(X)zIO/f(X)=f(CY)=d1y21+d2y22+dny2n.?g.E C?g.uE C?g.,kennef(X)=XTAX=nXi,j=1aijxixj C?g.,r(f)=r,K3_5O
8、 X=CY,f(X)z5/f(X)=f(CY)=y21+y22+y2r.yyydn,3_5O X=C1Z,f(X)zIO/.r(f)=r,f?IO/k r,”?c r,uIO/f(X)=f(C1Z)=d1z21+d2z22+drz2r=ZTZ,=diag(d1,d2,dr,0,0),CT1AC1=.?g.E C?g.-1=diag(1d1,1d2,1dr,1,1)KkT11=diag(1,1,0,0).u,2d_5O Z=1Y,Kkf(X)=f(C1Z)=f(C11Y)=(C11Y)TA(C11Y)=YT1CT1AC11Y=YTT11Y=YTdiag(1,1,0,0)Y=y21+y22+y2r
9、.1_?,C=C11_?.?L_5OX=CY f(X)z5/.?g.R?g.u R?g.,kennef(X)=XTAX=nXi,j=1aijxixj R?g.,r(f)=r,K3_5O X=CY,f(X)z5/f(X)=f(CY)=y21+y2p y2p+1 y2r.K?p d f(,=5/?.yyy,3_5O X=C1Z,f(X)zIO/.r(f)=r,f?IO/k r,”?c r,uIO/f(X)=f(C1Z)=d1z21+d2z22+drz2r=ZTZ,=diag(d1,d2,dr,0,0),CT1AC1=.”?d1,dp 0,dp+1,dr q.uy21+y2p y2p+1 y2r=w
10、21+w2q w2q+1 w2r(0.1)d X=CY=RW,?W=(R1C)Y.-R1C=(bij)nn,uw1=b11y1+b12y2+b1nynw2=b21y1+b22y2+b2nynwn=bn1y1+bn2y2+bnnyn(0.2)g5|b11y1+b12y2+b1nyn=0bq1y1+bq2y2+bqnyn=0yp+1=0yn=0?g.R?g.Tg5|k n,n (p q).p q,?u,Tg5|k).?(y1,yp,yp+1,yn)T=(c1,cp,0,0)T).)“(0.2),?(w1,wq,wq+1,wn)T=(0,0,bq+1,bn)T.“(0.1)?c21+c2p=b2q+
11、1 b2r c1,cp?,m,g.p=q.555333?ggg.?555/,p?.555,r p KKK.555,2p r?.?g.?g.?5O?g.A5,?nef(X)=XTAX=nXi,j=1aijxixj R?g.,K3?5O X=PY,f(X)zIO/f(X)=f(PY)=1y21+2y22+ny2n.1,2,n A?A?.?g.?5Oz?g.IO/?g.?A;?A?q?z.?g.f(x1,x2,x3,x4)=2x1x2+2x1x3 2x1x4 2x2x3+2x2x4+2x3x4?5OzIO/.)?g.f?A=0111101111011110.A?|E A|=?111111111111
12、?=(+3)(1)3?g.f?A?A?1=3,2=3=4=1.uA?1,)5|(1E A)X=0,?:)X1=(1,1,1,1)T z=?1=12(1,1,1,1)T.uA?2,)5|(2E A)X=0,?)x1=x2+x3 x4u?:)X2=(1,1,0,0)T,3=(1,0,1,0)T,4=(1,0,0,1)T.Schmidt?z=?2=12(1,1,0,0)T,3=16(1,1,2,0)T,4=123(1,1,1,3)T.L?5O X=(1,2,3,4)Y?g.zIO/.?g.f?5O-2x2+3y2+3z2 2xy 2xz=1zIO,L?-.)?g.f(x,y,z)=2x2+3y2+3
13、z2 2xy+2xzK?g.f?A=211130103.A?|E A|=?2111 3010 3?=(1)(3)(4).?g.f?A?A?1=1,2=3,3=4.A?i,i=1,2,3,)5|(iE A)X=0,O?:)X1=(2,1,1)T,2=(0,1,1)T,3=(1,1,1)T.=I z=1=16(1,1,1)T,2=12(0,1,1)T,3=13(1,1,1)T.L?5Oxyz=(1,2,3)uvw?g.zIO/f=u2+3v2+4w2.l?-zu2+3v2+4w2=1,d?.?g.4.7?g.?59?HO?g.?g.?5?R?g.f(X)=XTAX.eu Rn?,k f()=TA
14、0,K f?(positive definite)?g.?A?;eu Rn?,k f()=TA 0,f()0.=g(Y)E,?g.?g.A?g.f(X)=XTAX?(K)?(K).5?u n.?A?A?A?u.?1?u.?.?g.F?dAn SylvesterSSSfff?A=(aij)nn,?a11a12a1ka21a22a2k.ak1ak2akk?,k=1,2,n A?Sf.?g.F?dAn Sylvestern?g.f(X)=f(x1,x2,xn)=nXi,j=1aijxixj=XTAX?A?Sfu.yyy ky75.b?f?g.Pfk(x1,xk)=f(x1,xk,0,0)=kXi,j
15、=1aijxixj,k=1,2,n.fk?.du?g.?1?u,l?a11a12a1ka21a22a2k.ak1ak2akk?0,k=1,2,n?g.5?y2y5.n 8B.(1)?n=1,f(x1)=a11x21,d a11 0,w,f(x1)?.(2)b?(u n 1?g.ey n?/.PA=An1Tann!,C1=En1A1n101!KCT1AC1=En10T(A1n1)T1!An1Tann!En1A1n101!=An100ann TA1n1!?g.5?yd8Bb?,3 n 1?_?C2,?CT2An1C2=En1.-C=C1 C2001!,u C _,CTAC=CT2001!CT1AC
16、1 C2001!=CT2001!An100ann TA1n1!C2001!=En100ann TA1n1!du|A|0,?ann TA1n1 0.l?A?.?g.K?O?g.f(X)=f(x1,x2,xn)=nXi,j=1aijxixj=XTAX K?A?Sf?u,?Sfu.?g.f?g.f(x,y,z)=5x2+y2+5z2+4xy 8xz 4yz?5.)f(x,y,z)?524212425du a11=5 0,?a11a12a21a22?=?5221?=1 0,|A|=?524212425?=?101212425?=?100214429?=1 0.f(x,y,z)?g.?g.f?g.f(x,y,z)=5x2 6y2 4z2+4xy+4xz?5.)f(x,y,z)?522260204du a11=5 0,|A|=?522260204?=80 0,?a11a12a21a22?=?1tt4?=4 t2 0,|A|=?1t1t40102?=4 2t2 0.2 t 2.?g.