Skip to content

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

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
merged 4 commits into from
Jun 16, 2025
Merged

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

Changes from 3 commits
Commits
Show all changes
4 commits
Select commit Hold shift + click to select a range
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
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
4 changes: 2 additions & 2 deletions lectures/eigen_II.md
View file
Open in desktop
Original file line number Diff line number Diff line change
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 $k \in \mathbb{N}$.
Copy link

Copilot AI Apr 23, 2025

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

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

Copilot uses AI. Check for mistakes.

Copy link
Contributor

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

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

Copy link
Member

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

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],
Expand Down
Loading