第六章+范数及矩阵函数.doc

第六章范数及矩阵函数第六章范及矩阵函数数范的基本念数概?1范是更阵一般反映向量阵“距”的量。数离V(F)V(F)v定阵阵域数阵阵域或阵域~数数阵阵性空阵~阵到的FR映射~阵足,v(α)>0α;,1正定性阵中一切非零向量~有~Vα;,2阵次性阵中一切向量及中一切数~有VFkv(kα)=kv(α)~βv(α+β)=v(α)+v(β)α;,3三角不等式阵中一切向量~有~Vv阵称是上得范~阵范阵性空阵。数Vθ=0α,?v(θ)=0注,由。α=?α,α?是的一阵范。数VTnX=(x,x,,,x)例在中~有三阵常用的向量范~阵数1C12nnX=x范数~1—?1i=1i11n2H22X=(x)=(XX)范数~2—?i2=1iX=max{}xi?—数范?i1nppn?p?1~其中。是上的范。数?X=(x)CPp?iP=i111+=1p,q>1引理若阵数~且~阵阵一切正阵数有a,bpqpqabab?+。pq1p?1q?1p?1阵明如阵y=x?x=y=yybS2S1axOpaa1?pSxdx==~1?0pqbb1?qSydy==。2?0qS+S?ab而~得阵。12TX=(x,x,,,x)定理;不等式,阵~=(y,y,,,y)?C~阵12n11nnnnpqHpqYY=xy?xy?(x)(y)~????iiiiii=1=1=1=1iiii11++=1p,q?R其中~且。pq阵明第一不等式阵然成立~阵最后一不等式个个θ首先当和至少有一阵个阵~命阵成立。XYX?θ,Y?θ当阵~令xyiia,b==11iinnpqpq(x)(y)??ii=1=1iipqxyxyiiii?+11nnnn?pqpqpqpxqy??()()xyii??ii=1=1ii11==iipq,,11nnnxy??11pqiipq,,??+xy(x)(y)???iiiipqpq,,===xyiii111ii??,,11nnnpqpq?xy?(x)(y)???iiii===iii111p?1,x,y?C,定理;不等式,(x+y)?(x)+(y)???iiii===iii111p=1p>1阵明当阵~命阵成立。当阵nnpp(x+y)?(x+y)??iiii=1=1iinp?1=(x+y)(x+y)?iiiii1=nnp?1p?1=x(x+y)+y(x+y)??iiiiiii1i1==1111nnnn,,,,qqpp??pqppqp(1)(1)++?+(x)(xy)(y)(xy)????,,,,iiiiii,,,,iiii11==11==1,,?111nnn,,pppppp=++,,(x)(y)(xy)???,,iiii,,,ii,i111,,===1?111nnnp,,ppppp?(x+y)?(x)+(y)???iiii,,=1=1=1iii,,1npTnnp?X=(x,x,,,x)?C定理在中定阵~阵阵X=(x),?nip=1inlimX=XX1?p<+?~都是中的范~特阵数。Cpp?p?+?阵明正定性及阵次性阵然~下面阵三角不等式,x=θ当阵~命阵成立。x=max{}x?0x?θ当阵~令~于是ki111nnx1pipppp(x)=(())?n?1(p??)~??ixx=1=i1kik1pnxpi?limX=x=X()?1k而~?p?p?+?x=1iknn×n?X=AX例阵阵上的范~今有数~定阵2CA?CAn??detA?0阵明阵上的范数CA?X?θ,AX?θ?detA?0X=AX>0?detA?0;正定性,AkX=AkX=kAX=kX;阵次性,AAX+Y=A(X+Y)=AX+AYAAXAY?+=X+YAAε,ε,,,ε()VCVV定理阵阵阵阵性空阵~阵的一阵基~=xεX=(x,x,,,x)?C任一向量~。?12iin1n?ε=X?ξ?V?V又阵阵上的范~今定阵数~~阵阵上的Cvv范。数?ξ=X>0?ξ?θ?X?θ阵明;正定性,vkξ=kX=kX=kξ;阵次性,vvnTη=yεY=(y,y,,,y)~?12iin=1iξ+η=X+Y?X+Y=ξ+η。vvv?V定理阵阵阵性空阵上范~=0?α=θ,~1α?β?|α?β|,~2nε,ε,,,εξ=xεdimV=nV,阵~阵的一阵基~~阵312n?ii1ξx,x,,,x是的阵阵阵函。数n12nθ=0?α?θ=0α=0α=0阵明,由于。1α?θ?α>0α=0?α=θ若~故。α=α?β+β?α?β+β,~故2α?β?α?βα?β=β?α?β?α又α?β?|α?β|所以nx?a,i=1,2,,,n;α=aε,3?iiiii=1n0?|ξ?α|?ξ?α=(x?a)ε?iii=1inn:,?x?aε?εmax{}x?a?0