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n阶矩阵若存在正整数$$Ak?1α??=0，证明：向量组$<msup>\; <mrow></mrow>\; </msup>$

$\begin{array}{cc}{}_{}{}_{}{}_{}{}^{}0& \end{array}$

${}_{}{}^{}{}_{}{}^{}{}_{}{}^{}0$

$\begin{array}{cc}{}_{}{}_{}{}^{}{}_{}{}^{}0& \end{array}$λ3?=λ4?=?=λk?=0，从而得证$<msup>\; <mrow></mrow>\; </msup>$α,Aα,?,Ak?1α线性无关（这道题主要利用了构造方程求解）

$\begin{array}{cc}{}_{}{}_{}{}_{}{}^{}{}_{}{}^{}0& \end{array}$β=α1?+α2?+α3?+α4?利用特征向量的性质可得

$\begin{array}{cc}& {}_{}{}_{}{}_{}{}_{}{}_{}{}_{}{}_{}{}_{}{}_{}{}_{}{}_{}{}_{}{}_{}{}_{}\\ & {}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}{}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}0\end{array}$

$\begin{array}{cc}& {}_{}{}_{}{}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}{}_{}{}_{}{}_{}{}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}{}_{}\\ & {}_{}{}_{}{}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}{}_{}{}_{}{}_{}{}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}{}_{}0\end{array}$α1?,α2?,α3?,α4?分别属于不同的特征值，故线性无关从洏有

$\begin{array}{cc}\begin{array}{c}{}_{}{}_{}{}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}0\\ {}_{}{}_{}{}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}0\\ {}_{}{}_{}{}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}0\\ {}_{}{}_{}{}_{}{}_{}^{}{}_{}{}_{}^{}{}_{}\end{array}& \end{array}$k1?,k2?,k3?,k4?为待定的未知数，方程组$$4个方程组成的齐次线性方程组其系数行列式

$\begin{array}{cccc}& {}_{}& {}_{}^{}& {}_{}^{}\\ & {}_{}& {}_{}^{}& {}_{}^{}\\ & {}_{}& {}_{}^{}& {}_{}^{}\\ & {}_{}& {}_{}^{}& {}_{}^{}\end{array}$D=∣∣∣∣∣∣∣∣?1111?λ1?λ2?λ3?λ4??λ12?λ22?λ32?λ42??λ13?λ23?λ33?λ43??∣∣∣∣∣∣∣∣? λ1?,λ2?,λ3?,λ4?互异，所以

${}_{}{}_{}{}_{}{}_{}$β,Aβ,A2β,A3β线性无关（这道题主要利用了特征向量的性質求解）

B=??????β1?β2??βn????????,AB=C=??????γ1?γ2??γn????????，于是

$\begin{array}{cccc}{}_{}& {}_{}& & {}_{}\\ {}_{}& {}_{}& & {}_{}\\ & & & \\ {}_{}& {}_{}& & {}_{}\end{array}\begin{array}{c}{}_{}\\ {}_{}\\ \\ {}_{}\end{array}\begin{array}{c}{}_{}{}_{}{}_{}{}_{}{}_{}{}_{}\\ {}_{}{}_{}{}_{}{}_{}{}_{}{}_{}\\ \\ {}_{}{}_{}{}_{}{}_{}{}_{}{}_{}\end{array}\begin{array}{c}{}_{}\\ {}_{}\\ \\ {}_{}\end{array}$AB=??????a11?a21??a1m??a12?a22??a2m??????a1n?a2n??amn??????????????β1?β2??βn????????=??????a11?β1?+a12?β2?+?+a1n?βn?a21?β1?+a22?β2?+?+a2n?βn??a1m?β1?+a2m?β2?+?+amn?βn????????=??????γ1?γ2??γn????????, B的行向量线性表出故

α1?,α2?,?,αs?,β1?,β2?,?,βs?线性表出，故

$$A,B的列向量组的极大线性无关组分别为${}_{}{}_{}<msub></msub>$α1?,α2?,?,αp?扩充成$$[A,B]的极大线性无关组设为

[AEn??OB?]作初等变换，

$\begin{array}{cc}{}_{}& \\ & \end{array}{}_{}$r(AB)?r(A)+r(B)?n（这道题主要利用了矩阵的初等变换求解）

0

0 r(A?)=n。（或两边取行列式得${}^{}{}^{}{}^{}{}^{}{}^{}0{}^{}$

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