关于等价标准形的专题讨论

$f命题:$任意方阵$A$均可分解为可逆阵$B$与幂等阵$C$之积

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$f命题:$任意方阵$A$均可分解为可逆阵$B$与对称阵$C$之积

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$f命题:$设$A,B in {P^{n imes n}}$，且$rleft( A ight) + rleft( B ight) le n$，则存在$n$阶可逆阵$M$，使得$AMB = 0$

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$f命题:$设$A$为$n$阶方阵，则存在$n$阶方阵$B$，使得$A=ABA,B=BAB$

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$f命题:$设$A$为$n$阶矩阵且$ABA=A$有唯一解$B$,证明：$BAB=B$

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$f命题:$设$A$是秩为$r$的$m imes r$矩阵$left( {m > r} ight)$,$B$为$r imes s$矩阵,则存在可逆阵$P$,使得$PA$的后$m-r$行全为零

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$f命题:$设$T in Lleft( {V,n,F} ight)$,则存在$S in Lleft( {V,n,F} ight)$,使得$TST = T$

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$f命题:$设$A in {M_{m imes n}}left( F ight),B in {M_{n imes m}}left( F ight),m ge n,lambda e 0$，则

[{ m{ }}left| {lambda {E_m} - AB} ight| = {lambda ^{m - n}}left| {lambda {E_n} - BA} ight|]

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$f命题:$设$A,B,C$为$n$阶矩阵,且$AC=CB$,$rleft( C ight) = r$,证明：$A$与$B$至少有$r$个相同特征值

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$f命题:$设${alpha _1},{alpha _2}, cdots ,{alpha _n}$为${V_n}left( F ight)$的一个基,$A in {M_{n imes s}}left( F ight)$,且[left( {{eta _1},{eta _2}, cdots ,{eta _s}} ight) = left( {{alpha _1},{alpha _2}, cdots ,{alpha _n}} ight)A],证明：$dim Lleft( {{eta _1},{eta _2}, cdots ,{eta _s}} ight) = rleft( A ight)$

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$f命题:$设$A,B$为$n$阶矩阵，若$rleft( {AB} ight) = rleft( {BA} ight)$对任意的$B$成立，则$A = 0$或$A$可逆

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$f命题:$设$P in {F^{r imes m}},Q in {F^{n imes s}}$，若对任意的$A in {F^{m imes n}}$，都有$PAQ=0$，证明：$P=0或Q=0$

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$f命题:$设$A in {M_m}left( F ight),C in {M_n}left( F ight)$,若对于$B in {M_{mn}}left( F ight)$,有$rleft( {egin{array}{*{20}{c}}A&B \ 0&C end{array}} ight) = rleft( A ight) + rleft( C ight)$,证明：$A或C$可逆

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$f命题:$若$矩阵{A_{m imes n}}{B_{n imes p}}{C_{p imes q}}$的秩对一切秩$1$的矩阵$B$总为$1$，则$A$为列满秩，且$C$为行满秩

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$(04浙大七)$设$V = {P^{n imes n}}$看成数域$P$上的线性空间，取定$A,B,C,D in {P^{n imes n}}$，对任意$X in {P^{n imes n}}$，令[sigma left( X ight) = AXB + CX + XD]

证明：$(1)$$sigma $是$V$上的线性变换  $(2)$当$C = D = 0$时，$sigma $可逆的充要条件是$left| {AB} ight| e 0$

$(05浙大四)$设$A$为$n imes s$矩阵,证明：秩$(A)=r$的充要条件是存在两个列满秩的矩阵$B_{n imes r}$和$C_{s imes r}$，使得$A=BC^T$

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$(09江西师大)$设$A$为$m imes r$矩阵，证明：$A$列满秩的充要条件是存在$r imes m$的矩阵$B$，使得$BA=E_r$

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$(12川大)$设$A,B$为数域$F$上$m imes n$矩阵,证明：当$m e n$时,由$f(X)=AXB$给出的从${M_{n imes m}}left( F ight)$到${M_{m imes n}}left( F ight)$的线性映射$f$是不可逆的

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$(05川大七)$设$M_n(F)$是数域$F$上的$n$阶方阵全体,对任意非零矩阵$Ain M_n(F)$,定义集合${S_A} = left{ {XAY|forall X,Y in {M_n}left( F ight)} ight}$,证明：$S_A=M_n(F)$

$f命题:$

附录

$f命题:$设$A,B$为同型矩阵且$r(A)=r,r(B)=s$，证明：$r(A+B)=r(A)+r(B)$的充要条件是存在可逆阵$P,Q$，使得

[A = Pleft( {egin{array}{*{20}{c}}
{{E_r}}&0 \
0&0
end{array}} ight)Q，B = Pleft( {egin{array}{*{20}{c}}
0&0 \
0&{{E_s}}
end{array}} ight)Q]其中$r+s$不超过矩阵$A$的行数及列数

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$f命题:$设$G$为非零矩阵$A$的一个广义逆，即$AGA=A$，则存在可逆阵$P,Q$，使得

[A = Pleft( {egin{array}{*{20}{c}}
{{E_r}}&0 \
0&0
end{array}} ight)Q，G = {Q^{ - 1}}left( {egin{array}{*{20}{c}}
{{E_s}}&0 \
0&0
end{array}} ight){P^{ - 1}}]

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$f命题:$设$A in {M_n}left( F ight),rleft( A ight) = rleft( {{A^2}} ight)$，则存在可逆阵$P$，使得$A = Pleft( {egin{array}{*{20}{c}}D&0 \ 0&0 end{array}} ight){P^{ - 1}}$，其中$D$为可逆阵

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$f命题:$设$A,B$为$n$阶矩阵，且$BA=A$，$rleft( A ight) =rleft( B ight) $，则${B^2} = B$

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