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Studying for an exam and I came across an unfamiliar concept. Can anyone help me out with this practice question? I would really like if someone could go through the whole problem.

When $A$ = $ \{ \begin{bmatrix} -1 \ -2 \ \quad \space 2 \\ -1 \ -1 \ \space -9 \\ \quad 0 \ -1 \ -9 \end{bmatrix} \} $ , any sequence $ \quad \{ A^k \vec{x}$₀$ \} \quad $ is dominated by a scalar multiple of the sequence $ \quad \{ ( \lambda^k) \vec{v} \} \quad $ for some value $\space \lambda \space $ and some vector $ \space \vec{v} \space $ as $ \space k \space $ gets large. What are $ \lambda $ and $ \vec{v} \space $?

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1 Answer

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$\lambda$ is the largest (absolute value) characteristic value of $A$ and $v$ the associated eigenvector. In the special case (such as we have with this matrix) that the largest absolute value eigenvalue is distinct from the other eigenvalues, we can use the power method to find this eigenvalue, and its associated eigenvector. Here is the wikipedia link: Power Method on Wikipedia

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