Let c 1, c2 is the different eigenvalues of two A's, and X and Y are their corresponding eigenvectors respectively, where

A * x = c 1 * x

A * y = c2 * y

Take transpose respectively, and multiply both sides by y and x respectively.

x' * A' * y = c 1 * x' * y

y' * A' * x = c2 * y' * x = c2 * x' * y

Correspondence subtraction

(c 1-C2)x ' * y = x ' * A ' * y-y ' * A ' * x = 0

And c 1-c2 ≠ 0, so x' * y = 0.

Certificate of completion