设A为n*m矩阵,B为m*n矩阵,如果E-AB可逆,证E-BA也可逆,并求出（E-BA）^-1

Question

设A为n*m矩阵,B为m*n矩阵,如果E-AB可逆,证E-BA也可逆,并求出（E-BA）^-1

Answer 1

证:因为
(E-BA)[E+B(E-AB)^-1A]
= E-BA+B(E-AB)^-1A-BAB(E-AB)^-1A
= E-BA+B(E-AB)(E-AB)^-1A
= E-BA+BA
= E.
所以 E-BA 可逆,且 (E-BA)^-1 = E+B(E-AB)^-1A.