Fix definition of primitive matrix and clarify Hamilton matrix reference

[Nanakjoth]

This PR fixes two minor issues referenced in #576:

1. Primitive matrix definition : Updated the condition to say $A^k$ is positive for some $k \in \mathbb{N}$, which is the correct definition of a primitive matrix.

2. Hamilton matrix: Replaced the misleading term "Hamilton matrix" with "Hamilton’s transition matrix (mc_eg2)" to avoid confusion with Hamiltonian matrices in physics.

I hope this fix was helpful, Happy to make any changes if needed — just let me know.

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[HumphreyYang]

Thanks @Nanakjoth for contribution! I just pushed some minor updates to the PR.

[Nanakjoth]

The update to use an internal {ref} ensures consistency in documentation referencing.

[mmcky]

thanks @Nanakjoth and @HumphreyYang

@Nanakjoth here is the documentation for the syntax on internal references for jupyter-book
https://jupyterbook.org/en/stable/content/references.html.

[Copilot]

Pull Request Overview

This pull request refines the definition of a primitive matrix and clarifies the reference to Hamilton’s transition matrix.

• Updates the primitive matrix condition to state that A^k is everywhere positive for some k.
• Changes the reference for the Hamilton matrix to use a reStructuredText reference pointing to mc_eg2.

Comments suppressed due to low confidence (1)

lectures/eigen_II.md:395

• [nitpick] To fully align with the PR description and avoid any ambiguity with Hamiltonian matrices from physics, consider updating the reference text to explicitly say 'Hamilton’s transition matrix'.

So Perron-Frobenius theorem explains why both {ref}`Imam and Temple matrix <mc_eg3>` and {ref}`Hamilton matrix <mc_eg2>` converge to a stationary distribution, which is the Perron projection of the two matrices

[Copilot]

[nitpick] The updated definition appears correct; however, consider explicitly stating 'strictly positive' to align precisely with the standard mathematical definition of a primitive matrix.

Suggested change

	$A$ here is also a primitive matrix since $A^k$ is everywhere positive for some $k \in \mathbb{N}$.
	$A$ here is also a primitive matrix since $A^k$ is strictly positive for some $k \in \mathbb{N}$.

[mmcky]

@HumphreyYang @jstac what do you think of this [nitpick] AI review?

[HumphreyYang]

Many thanks @mmcky,

I think everywhere positive matches the definition of primitive matrices so I think we should keep it : )

[Nanakjoth]

Thanks for sharing the documentation, @mmcky! 📚, Now after the AI review, it looks Great. Please let me know if there’s anything else I can help with, I’m looking forward to seeing this merged😊.

[mmcky]

thanks so much @Nanakjoth and @HumphreyYang