Hi! This might be a silly question, but I can't seem to figure it out and have not found any remarks on it in the literature. When diagonalizing an NxN matrix A, we solve the characteristic equation: Det(A - mI) = 0 which gives us the N eigenvalues m. Then, to find the eigenvectors v of A, we solve the eigenvalue problem Av = mv Now, any scalar multiple of an eigenvector of A is itself an eigenvector with the same eigenvalue. So, which eigenvector do we find when solving the eigenvalue problem? It can't be totally random, can it? Is there a way of determining which of the infinitude of eigenvectors (all with the same "direction") the algorithm chooses?