Right eigenvector λ and left eigenvector are orthogonal.
Let x be a (right) eigenvector of A corresponding to an eigenvalue λ and let y be a left eigenvector of A corresponding to a different eigenvalue µ, where λ $\neq$ µ. Show that x∗y = 0. Hint : Ax = λx and y'A = µy'
Solution 1:
- Step 1) $Ax=λx$
- Step 2) $y'Ax=λy'x$
- Step 3) $y'Ax-λy'x=0$
- Step 4) $(y'A-λy')x=0$
- Step 5) $(\mu y'-λy')x=0$
- Step 6) $(\mu-λ)y'x=0$
- How: $\mu \neq λ \implies \mu-λ \neq 0$
- This way: $y'x=0$
Solution 2:
Hint: Calculate $y'Ax$ two different ways, and relate the answer to $y'x$.