I just have a quick question to the following approximation.

Let $A \in \mathbb{R}^{n \times n} $ be a primitive Matrix with Eigenvalues $ \lambda_1,\dots,\lambda_n $ ordered this way, $ \lambda_1 > | \lambda_2 | \geq \cdots \geq | \lambda_n |$.

Then we receive,

$A^k = \lambda_1 u_1\cdot v_1^T + \mathcal{O}(k^{m_2-1}|\lambda_2|^k)$,

where $ u_1, v_1 $ are Right- and Lefteigenvector with Eigenvalue $\lambda_1$. If $ |\lambda_2|=\cdots = |\lambda_j| $ for some $ j \geq 3$ then $m_2 \geq m_j $ where $ m_j$ is the algebraic multiplicity of the Eigenvalue $ \lambda_j$ .

I am not sure if we mean by $ m_2 $ the algebraic multiplicity of $\lambda_2$ and why we say that $m_2 \geq m_j $. Additonally I don't know how we are getting $ \mathcal{O}(k^{m_2-1}|\lambda_2|^k)$. If $ m_2 $ is the algebraic multiplicity of $\lambda_2$ then I understand where the Landau comes from, as this is related to the convergence of a Jordanblock with length $m_2$.


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