设a为n阶矩阵,证明存在一可逆矩阵b及一幂等矩阵c(c=c^2),使a=bc
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设A为n阶方阵,证明存在一可逆矩阵B及一幂等矩阵C,使A等于BC答:幂等矩阵定义是 C^2=C 设A的标准型为F= E 0 0 0 即可设A=PFQ,其中P,Q可逆,A=PQQ^{-1}FQ,令B=PQ,B可逆,且令C=Q^{-1}FQ,由于F^2=F,所以C^2=C.
设A为n阶幂等矩阵,秩为r,证明存在矩阵B,C,使A=CB,且BC=I,B,C秩均为r答:首先对A做满秩分解A=CB,然后C(BCB)=A^2=A=CB,所以C(BCB-B)=0 注意C的列线性无关,得到BCB=B,再利用B的行线性无关得到BC=I_r