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Fix definition of primitive matrix and clarify Hamilton matrix reference #584

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Fix definition of primitive matrix and clarify Hamilton matrix reference #584

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c41a6da
Fix definition of primitive matrix and clarify Hamilton matrix reference
Nanakjoth Apr 21, 2025
5bcfbf2
minor updates
HumphreyYang Apr 21, 2025
b071ffe
remove spaces
HumphreyYang Apr 21, 2025
19602c3
Merge branch 'main' into fix-primitive-matrix-typos
HumphreyYang Apr 24, 2025
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4 changes: 2 additions & 2 deletions lectures/eigen_II.md
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Expand Up @@ -238,7 +238,7 @@ A = \begin{bmatrix} 0.5 & 0.1 \\
\end{bmatrix}
$$

$A$ here is also a primitive matrix since $A^k$ is everywhere nonnegative for $k \in \mathbb{N}$.
$A$ here is also a primitive matrix since $A^k$ is everywhere positive for some $k \in \mathbb{N}$.
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Copilot AI Apr 23, 2025

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

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$A$ 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 $k \in \mathbb{N}$.

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

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Many thanks @mmcky,

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


$$
B = \begin{bmatrix} 0 & 1 \\
Expand Down Expand Up @@ -392,7 +392,7 @@ We are now prepared to bridge the languages spoken in the two lectures.

A primitive matrix is both irreducible and aperiodic.

So Perron-Frobenius theorem explains why both {ref}`Imam and Temple matrix <mc_eg3>` and [Hamilton matrix](https://en.wikipedia.org/wiki/Hamiltonian_matrix) converge to a stationary distribution, which is the Perron projection of the two matrices
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

```{code-cell} ipython3
P = np.array([[0.68, 0.12, 0.20],
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