How to prove two eigenvectors corresponding to the same eigenvalue of a matrix cannot be orthogonal to each other.

Question

First, let x, y be the right and left eigenvectors

let λ1 and λ2 be different eigenvalues

So, I can consider Ax=λ1x and then I will find the result and (λ2-λ1)y*x=0

Since λ2 is not equal to λ1, y*x=0 => Orthogonal

However, how to prove two eigenvectors corresponding to the same eigenvalue of a matrix cannot be orthogonal to each other?

Answer 1

This is not true. Take the identity matrix. Every vector is an eigen vector corresponding to eigen value $1$!

Answer 2

It is false. Take for instance the identity matrix. Every vector is an eigenvector, and you can choose lots and lots of orthogonal pairs of vectors.