若尔当标准形的几何理论(1)
找一组基使线性变换
在这组基下的矩阵称为若尔当标准形
定义：对于线性空间V中的线性变换的多项式)
及任意向量
,若有%5Cvarepsilon%3D0)
,则称)
是对于的零化多项式,若是对于的零化多项式中次数最低的首一多项式,则称为对于的最小多项式
易证对的最小多项式整除对的任一零化多项式
引理：对
上有限维空间
上的线性变换,下列结论等价
1.在基
下的矩阵是若尔当块
%3D%5Cbegin%7Bpmatrix%7D%5Clambda_0%260%260%26%5Ccdots%260%260%260%5C%5C%201%26%5Clambda_0%260%26%5Ccdots%260%260%260%5C%5C%20%5Cvdots%26%5Cvdots%26%5Cvdots%26%20%26%5Cvdots%26%5Cvdots%26%5Cvdots%5C%5C%200%260%260%26%5Ccdots%261%26%5Clambda_0%260%5C%5C%200%260%260%26%5Ccdots%260%261%26%5Clambda_0%5Cend%7Bpmatrix%7D%E2%80%8B)
2.
,%5Cvarepsilon_0)
,%5E2%5Cvarepsilon_0)
,
,%5E%7Bn-1%7D%5Cvarepsilon_0)
是的基且%5En%5Cvarepsilon_0%3D(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%5Cvarepsilon_%7Bn-1%7D%3D0)
3.![V=P[\mathscr{A}]\varepsilon_0=\{f(\mathscr{A}\varepsilon_0|f(\lambda)\in\C[\lambda]\}](https://math.jianshu.com/math?formula=V%3DP%5B%5Cmathscr%7BA%7D%5D%5Cvarepsilon_0%3D%5C%7Bf(%5Cmathscr%7BA%7D%5Cvarepsilon_0%7Cf(%5Clambda)%5Cin%5CC%5B%5Clambda%5D%5C%7D)
,且%5En)
是的最小多项式
证明：
由线性变换矩阵的定义,显然成立
必要性
,有
%5Cvarepsilon_0%2B%5Ccdots%2Bl_%7Bn-1%7D(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%5E%7Bn-1%7D%5Cvarepsilon_0)
![=l_0\mathscr{E}+l_1(\mathscr{A}-\lambda_0\mathscr{E})+\cdots+l_{n-1}(\mathscr{A}-\lambda_0\mathscr{E})\varepsilon_0\in P[\mathscr{A}]\varepsilon_0](https://math.jianshu.com/math?formula=%3Dl_0%5Cmathscr%7BE%7D%2Bl_1(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%2B%5Ccdots%2Bl_%7Bn-1%7D(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%5Cvarepsilon_0%5Cin%20P%5B%5Cmathscr%7BA%7D%5D%5Cvarepsilon_0)
此时是的一个零化多项式
设为%3Dl_0%2Bl_1(%5Clambda-%5Clambda_0)%2B%5Ccdots%2Bl_%7Bn-1%7D(%5Clambda-%5Clambda_0)%5E%7Bn-1%7D)
由%5Cvarepsilon_0%3D0)
![[l_0\mathscr{E}+l_1(\mathscr{A}-\lambda_0\mathscr{E})+\cdots+l_{n-1}(\mathscr{A}-\lambda_0\mathscr{E})^{n-1}]\varepsilon_0](https://math.jianshu.com/math?formula=%5Bl_0%5Cmathscr%7BE%7D%2Bl_1(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%2B%5Ccdots%2Bl_%7Bn-1%7D(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%5E%7Bn-1%7D%5D%5Cvarepsilon_0)
%5Cvarepsilon_0%2B%5Ccdots%2Bl_%7Bn-1%7D(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%5E%7Bn-1%7D%5Cvarepsilon_0%3D0)
但%5Cvarepsilon_0%2C%5Ccdots%2C(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%5E%7Bn-1%7D%5Cvarepsilon_0)
是的一组基,线性无关
故
即%3D0)
故是的最小多项式
充分性
首先是的零化多项式
故%5En%5Cvarepsilon_0%3D0)
![\forall v\in V=P[\mathscr{A}]\varepsilon_0](https://math.jianshu.com/math?formula=%5Cforall%20v%5Cin%20V%3DP%5B%5Cmathscr%7BA%7D%5D%5Cvarepsilon_0)
有![f(\lambda)\in P[\lambda],v=f(\mathscr{A})\varepsilon_0](https://math.jianshu.com/math?formula=f(%5Clambda)%5Cin%20P%5B%5Clambda%5D%2Cv%3Df(%5Cmathscr%7BA%7D)%5Cvarepsilon_0)
作带余除法,%3D(%5Clambda-%5Clambda_0)%5Eng(%5Clambda)%2Bl_0%2Bl_1(%5Clambda-%5Clambda_0)%2B%5Ccdots%2Bl_%7Bn-1%7D(%5Clambda-%5Clambda_0)%5E%7Bn-1%7D)
则有%5Cvarepsilon_0%3Dg(%5Cmathscr%7BA%7D)(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%5En%5Cvarepsilon_0%2Bl_0%5Cvarepsilon_0%2Bl_1(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%5Cvarepsilon_0%2B%5Ccdots%2Bl_%7Bn-1%7D(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%5E%7Bn-1%7D%5Cvarepsilon_0)
%5Cvarepsilon_0%2B%5Ccdots%2Bl_%7Bn-1%7D(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%5E%7Bn-1%7D%5Cvarepsilon_0)
即为%5Cvarepsilon_0%2C%5Ccdots%2C(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%5E%7Bn-1%7D%5Cvarepsilon_0%E2%80%8B)
的线性组合
设%5Cvarepsilon_0%2B%5Ccdots%2Bl_%7Bn-1%7D(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%5E%7Bn-1%7D%5Cvarepsilon_0%3D0)
则![[l_0\mathscr{E}+l_1(\mathscr{A}-\lambda_0\mathscr{E})+\cdots+l_{n-1}(\mathscr{A}-\lambda_0\mathscr{E})]\varepsilon_0=0](https://math.jianshu.com/math?formula=%5Bl_0%5Cmathscr%7BE%7D%2Bl_1(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%2B%5Ccdots%2Bl_%7Bn-1%7D(%5Cmathscr%7BA%7D-%5Clambda_0%5Cmathscr%7BE%7D)%5D%5Cvarepsilon_0%3D0)
令%3Dl_0%2Bl_1(%5Clambda-%5Clambda_0)%2B%5Ccdots%2Bl_%7Bn-1%7D(%5Clambda-%5Clambda_0)%5E%7Bn-1%7D)
则%5Cvarepsilon_0%3D0)
若%5Cneq%200)
,则)%5Cle%20n-1)
与是的最小多项式矛盾
故%3D0)
故
即证线性无关
故为的基
定理：,如上
则在某基下的矩阵为若尔当形
%5C%5C%26J(%5Clambda_2%2Ck_2)%5C%5C%26%20%26%5Cddots%5C%5C%26%20%26%20%26J(%5Clambda_s%2Ck_s)%5Cend%7Bpmatrix%7D%E2%80%8B)
的充要条件为中存在
,使
![V=P[\mathscr{A}]\eta_1\oplus P[\mathscr{A}]\eta_2\oplus \cdots\oplus P[\mathscr{A}]\eta_s](https://math.jianshu.com/math?formula=V%3DP%5B%5Cmathscr%7BA%7D%5D%5Ceta_1%5Coplus%20P%5B%5Cmathscr%7BA%7D%5D%5Ceta_2%5Coplus%20%5Ccdots%5Coplus%20P%5B%5Cmathscr%7BA%7D%5D%5Ceta_s)
且每个
的最小多项式是%5E%7Bk_i%7D)
证明：
%5C%5C%26J(%5Clambda_2%2Ck_2)%5C%5C%26%20%26%5Cddots%5C%5C%26%20%26%20%26J(%5Clambda_s%2Ck_s)%5Cend%7Bpmatrix%7D)
是-不变子空间的直和
且每个
在
上有基使它的矩阵是)
,对每个
,有
使![W_i=P[\mathscr{A}]\eta_i](https://math.jianshu.com/math?formula=W_i%3DP%5B%5Cmathscr%7BA%7D%5D%5Ceta_i)
且对的最小多项式为%5E%7Bk_i%7D%2Ci%3D1%2C2%2C%5Ccdots%2Cs%5Cqquad%5Cmathcal%7BQ.E.D%7D%E2%80%8B)
注：定理说明,要证若尔当标准形存在,只需证存在不变子空间的直和分解
