Question

Show that eigen vectors of matrix and inverse matrices are same

Answer 1

Answer:

As we are interested in matrices with an inverse, the eigenvalues are nonzero.

v is an eigenvector of A corresponding to the eigenvalue λ≠0

=> Av = λv

=> A⁻¹Av = A⁻¹λv,  

=> v = λA⁻¹v

=> λ⁻¹v = A⁻¹v

=> v is an eigenvector of A⁻¹ corresponding to the eigenvalue 1 / λ