Question

Let V be a finite dimensional vector space over the field of complex numbers (C), and let T be an invertible linear operator on V. Prove that if c doesn't equal 0 is an eiganvalue of T, then 1/c is an eiganvalue of T^-1

Answer 1

Suppose \(x\) is the eigen vector of \(T\) corresponds to the eigne value \(c\).
Then \(T(x)=cx\)
Now \(x=T^{-1}(T(x))=T^{-1}(cx)=cT^{-1}(x)\)
Therefore \(T^{-1}(x)=\frac{1}{c} x\).