(a)If λ is a Eigen value of T then, ∃v(=0)∈V∋Tv=λvNow as T is invertible, Tv=λv⇒T−1(T(v))=T−1(λv)=λT−1(v)⇒T−1(v)=λ1vHence 1/λ is the eigen value of T−1.And v is also the Eigen vector of T−1.(i)
Conversely, If 1/λ is a Eigen value of T−1 then, ∃w(=0)∈V∋T−1w=w/λNow as T is invertible, T−1w=w/λ⇒T(T−1(w))=T(w)/λ⇒T(w)=λwHence λ is the eigen value of T.And w is also the Eigen vector of T. (ii)
(b)From (i) and (ii),We can see that both T and T−1 have same eigen vectors.
Comments