Cayley-Hamilton定理(凯莱-哈密顿定理)

1. 定义
(1) 符号定义

单位矩阵为 I I I,矩阵 A A A的行列式记作 det ⁡ ( A ) \det \left( A \right) det(A),伴随矩阵记作 a d j ( A ) \mathrm{adj} \left(A\right) adj(A).

(2) 特征多项式

矩阵 A A A的特征多项式定义为:
χ A ( s ) ≜ det ⁡ ( s I − A ) = s n + d 1 s n − 1 + ⋯ + d n , \chi _A\left( s \right) \triangleq \det \left( sI-A \right) =s^n+d_1s^{n-1}+\cdots +d_n, χA(s)det(sIA)=sn+d1sn1++dn,

2. 定理内容

χ A ( A ) = A n + d 1 A n − 1 + ⋯ + d n = 0 \chi _A\left( A \right) =A^n+d_1A^{n-1}+\cdots +d_n=0 χA(A)=An+d1An1++dn=0

3. 证明

考虑矩阵 ( s I − A ) \left( sI-A \right) (sIA)的逆,可以表示为:
( s I − A ) − 1 =    a d j ( s I − A ) det ⁡ ( s I − A ) ⋯ ( ∗ ) , \left( sI-A \right) ^{-1}=\frac{\,\,\mathrm{adj}\left( sI-A \right)}{\det \left( sI-A \right)}\cdots \left( * \right) , (sIA)1=det(sIA)adj(sIA)()

其中 ( s I − A ) \left( sI-A \right) (sIA)的行列式可以表示为 { 1 , s , s 2 , ⋯   , s n } \left\{ 1,s,s^2,\cdots ,s^n \right\} {1,s,s2,,sn}的线性组合,即
det ⁡ ( s I − A ) = s n + d 1 s n − 1 + ⋯ + d n . \det \left( sI-A \right) =s^n+d_1s^{n-1}+\cdots +d_n. det(sIA)=sn+d1sn1++dn.

( s I − A ) \left( sI-A \right) (sIA)的伴随矩阵可以表示为 { 1 , s , s 2 , ⋯   , s n − 1 } \left\{ 1,s,s^2,\cdots ,s^{n-1} \right\} {1,s,s2,,sn1}的线性组合,即
a d j ( s I − A ) = B 0 s n − 1 + B 1 s n − 2 + ⋯ + B n − 1 . \mathrm{adj}\left( sI-A \right) =B_0s^{n-1}+B_1s^{n-2}+\cdots +B_{n-1}. adj(sIA)=B0sn1+B1sn2++Bn1.(注:根据伴随矩阵的定义,可以知道多项式最高阶数为 ( n − 1 ) (n-1) (n1)

下证:
χ A ( A ) = A n + d 1 A n − 1 + ⋯ + d n = 0. \chi _A\left( A \right) =A^n+d_1A^{n-1}+\cdots +d_n=0. χA(A)=An+d1An1++dn=0.
( ∗ ) (*) ()式,在等号两边右乘 ( s I − A ) \left( sI-A \right) (sIA),左乘
det ⁡ ( s I − A ) I \det \left( sI-A \right) I det(sIA)I,可以得到
a d j ( s I − A ) ( s I − A ) = det ⁡ ( s I − A ) I . \mathrm{adj}\left( sI-A \right) \left( sI-A \right) =\det \left( sI-A \right) I. adj(sIA)(sIA)=det(sIA)I.

对等号左边进行展开
L H S = ( B 0 s n − 1 + B 1 s n − 2 + ⋯ + B n − 1 ) ( s I − A ) = ( B 0 s n + B 1 s n − 1 + ⋯ + B n − 1 s ) − ( B 0 A s n − 1 + B 1 A s n − 2 + ⋯ + B n − 1 A ) = B 0 s n + ( B 1 − B 0 A ) s n − 1 + ⋯ + ( B n − 1 − B n − 2 A ) s − B n − 1 A . \begin{aligned} \mathrm{LHS}&=\left( B_0s^{n-1}+B_1s^{n-2}+\cdots +B_{n-1} \right) \left( sI-A \right) \\ &=\left( B_0s^n+B_1s^{n-1}+\cdots +B_{n-1}s \right) -\left( B_0As^{n-1}+B_1As^{n-2}+\cdots +B_{n-1}A \right) \\ &=B_0s^n+\left( B_1-B_0A \right) s^{n-1}+\cdots +\left( B_{n-1}-B_{n-2}A \right) s-B_{n-1}A. \end{aligned} LHS=(B0sn1+B1sn2++Bn1)(sIA)=(B0sn+B1sn1++Bn1s)(B0Asn1+B1Asn2++Bn1A)=B0sn+(B1B0A)sn1++(Bn1Bn2A)sBn1A.

而等式右边为
R H S = s n I + d 1 s n − 1 I + ⋯ + d n I . \mathrm{RHS}=s^nI+d_1s^{n-1}I+\cdots +d_nI. RHS=snI+d1sn1I++dnI.

对照系数,有
{ B 0 = I B 1 − B 0 A = d 1 I ⋮ B n − 1 − B n − 2 A = d n − 1 I O − B n − 1 A = d n I ⇒ { B 0 A n = A n B 1 A n − 1 − B 0 A n − 2 = d 1 A n − 1 ⋮ B n − 1 A − B n − 2 A 2 = d n − 1 A − B n − 1 A = d n I \begin{cases} B_0=I\\ B_1-B_0A=d_1I\\ \vdots\\ B_{n-1}-B_{n-2}A=d_{n-1}I\\ O-B_{n-1}A=d_nI\\ \end{cases}\Rightarrow \begin{cases} B_0A^n=A^n\\ B_1A^{n-1}-B_0A^{n-2}=d_1A^{n-1}\\ \vdots\\ B_{n-1}A-B_{n-2}A^2=d_{n-1}A\\ -B_{n-1}A=d_nI\\ \end{cases} B0=IB1B0A=d1IBn1Bn2A=dn1IOBn1A=dnI B0An=AnB1An1B0An2=d1An1Bn1ABn2A2=dn1ABn1A=dnI

上式等号左边累加结果为 O O O(零矩阵),而右边累加为 A A A的特征多项式 χ A ( A ) \chi _A\left( A \right) χA(A),得证.

3. 应用

TODO

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