Convergence of QR method for eigenvalues

Question

Let $A = A^{(0)} \in \mathbb{M}_n(\mathbb{R})$ be a real square matrix. Suppose that $A$ is diagonalizable with eigenvalues: $\lambda_1,...,\lambda_n$. The QR method for calculating eigenvalues of A is described as follows: For each $k \in \mathbb{N}^{*}$:

1. Form the QR factorization of $A^{(k-1)}: A^{(k-1)} = Q^kR^k$.
2. Put $A^{k} = R^kQ^k$

Ander some assumptions on A, this method converges to a triangular matrix having the same eigenvalues as $A$. My problem is with those assumptions, I find them different from reference to an other, the different assumptions I have found are:

• $|\lambda_1|<|\lambda_2| < \cdots < |\lambda_n| $.
• The principal minors of $A$ don't vanish.
• $ A = P^{-1}DP $ such that $P$ has a LU factorisation. ($D$ is a diagonal matrix).

Which one is the right assumption ?