This seems a simple question but I can't find the solution by myself. Please help. Say we have a 2 by 2 matrix with different eigenvalues. Corresponding to each eigenvalue, there are a number of eigenvectors. Q1. Could the eigenvectors corresponding to the same eigenvalue have different directions? Q2. Could the eigvenvectors corresponding to the same eigenvalue be orthogonal? Q3. How can we prove that there is a pair of orthogonal eigenvalues for the 2 by 2 matrix, each for one eigenvalue? Your help would be appreciated.