关于幂等阵与幂幺阵的专题讨论

幂等阵

$f命题：$设$n$阶幂等阵$A$满足$A=A_{1}+cdots+A_{s}$，且$$r(A)=r(A_{1})+cdots+r(A_{s})$$

证明：所有的$A_{i}$都相似于一个对角阵，且$A_{i}$的特征值之和等于$A_{i}$的秩

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$f命题：$

$f(10北科大八)$设线性空间$V$的线性变换$sigma$满足${sigma ^2} = sigma $，证明：

(1)$V$中向量$eta$属于$sigma$的像集$ ext{Im}sigma$当且仅当$sigma (eta)=eta$

(2)$V = operatorname{Im} sigma  oplus { ext{Ker}}sigma $，且$V$的任一向量的直和分解为$alpha  = sigma left( alpha   ight) + left( {alpha  - sigma left( alpha   ight)} ight)$

(3)对任一直和分解$V = {V_1} oplus {V_2}$，存在唯一的幂等变换$sigma$，使得${V_1} = operatorname{Im} sigma ,{V_2}{ ext{ = Ker}}sigma $

(4)每个幂等变换都有方阵表示$left( {egin{array}{*{20}{c}}E&0 \ 0&0 end{array}} ight)$

幂幺阵

$f命题：$设$A$为$n$阶对合阵，即${A^2} = E$，则存在正交阵$Q$，使得${Q^{ - 1}}AQ = left( {egin{array}{*{20}{c}}{{E_r}}&0 \ 0&{ - {E_{n - r}}}end{array}} ight)$ 

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$f命题：$设${A^n} = {E_m}$，则$(E-A)x=0$的解空间的维数为$frac{1}{n}trleft( {A + {A^2} +  cdots  + {A^n}} ight)$

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$f命题：$

$f(06中科院)$设$f$为有限维向量空间$V$上的线性变换，且$f^n$是$V$上的恒等变换，这里$n$是某个正整数，设$W = { v in V|f(v) = v} $，证明：$W$是$V$的一个子空间，且其维数等于线性变换$left( {f + {f^2} +  cdots  + {f^n}} ight)/n$的迹

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附录1(幂等阵)

$f定义：$设$A$为$n$阶矩阵，若${A^2} = A$，则称$A$为幂等阵

$f命题1：$若$A$为幂等阵，则${A^T},{A^k},E - A$均为幂等阵

 

$f命题2：$幂等阵的特征值与行列式只能是$0$或$1$

 

$f命题3：$设$A$是特征值全为$0$或$1$的方阵，则$A$为幂等阵的充要条件是$A$可对角化

 

$f命题4：$$A$为幂等阵当且仅当$rleft( A ight) + rleft( {E - A} ight) = n$

 

$f命题5：$$A$为幂等阵当且仅当${F^n} = Nleft( A ight) oplus Nleft( {E - A} ight)$

 

$f命题6：$$A$为幂等阵当且仅当存在可逆阵$P$，使得${P^{ - 1}}AP = left( {egin{array}{*{20}{c}}{{E_r}}&0\0&0end{array}} ight),r = rleft( A ight)$

 

$f命题7：$设$A$为秩为$r$的幂等阵，则$trleft( A ight) = rleft( A ight)$

 

$f命题8：$设$A$为秩为$r$的幂等阵，则$left| {aE + bA} ight| = {left( {a + b} ight)^r}{a^{n - r}}$

 

$f命题9：$任意幂等阵均可分解为对称阵与正定阵之积

$f命题10：$设${A_1}, cdots ,{A_k}$均为$n$阶矩阵，$A = sumlimits_{i = 1}^k {{A_i}} $，则如下4条件中：

[left( 1 ight),left( 2 ight) Leftrightarrow left( 1 ight),left( 3 ight) Leftrightarrow left( 3 ight),left( 4 ight) Leftrightarrow left( 2 ight),left( 3 ight)]

$left( 1 ight){A_i}^2 = {A_i}left( {i = 1,2, cdots ,k} ight)$

$left( 2 ight){A_i}{A_j} = 0left( {i e j} ight),rleft( {{A_i}^2} ight) = rleft( {{A_i}} ight)$

$left( 3 ight){A^2} = A$

$left( 4 ight)rleft( A ight) = sumlimits_{i = 1}^k {rleft( {{A_i}} ight)} $

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