设α=[1,0,1]T,A=ααT,n是正数,则|aE-An|=______

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问题 设α=[1,0,1]T,A=ααT,n是正数,则|aE-An|=______

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答案a2(a-2n)

解析 A=ααT=
An=(ααT)n=ααTααT…αTα=α(αTα)(αTα)…(αTα)αT=2n-1A,

=a[(a-2n-1)2-(-2n-1)2]
=a(a2-a2n)=a2(a-2n).
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