用A'表示A的转置,E表示单位阵.
由A为正交阵,有A'A = E.
于是|E-A| = |A'A-A|
= |(A'-E)A|
= |A'-E|·|A|
= |A'-E| (∵|A| = 1)
= |(A-E)'| (∵E' = E)
= |A-E|
= |-(E-A)|
= (-1)^(2n+1)·|E-A|
= -|E-A|.
因此|E-A| = 0,即1是A的一个特征值.