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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?

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# Diagonalization, which eigenvector is found?

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