From c41a6daf064d74aa3712620059b495a9ea47e9e4 Mon Sep 17 00:00:00 2001 From: Nanakjoth Date: Mon, 21 Apr 2025 12:37:00 +0530 Subject: [PATCH 1/3] Fix definition of primitive matrix and clarify Hamilton matrix reference --- lectures/eigen_II.md | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/lectures/eigen_II.md b/lectures/eigen_II.md index 5ccd9ebb..d78dca15 100644 --- a/lectures/eigen_II.md +++ b/lectures/eigen_II.md @@ -238,7 +238,8 @@ A = \begin{bmatrix} 0.5 & 0.1 \\ \end{bmatrix} $$ -$A$ here is also a primitive matrix since $A^k$ is everywhere nonnegative for $k \in \mathbb{N}$. +A here is also a primitive matrix since $A^k$ is everywhere positive for some $k \in \mathbb{N}$. + $$ B = \begin{bmatrix} 0 & 1 \\ @@ -392,7 +393,8 @@ 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 ` 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 the Perron-Frobenius theorem explains why both the Imam and Temple matrix and Hamilton’s transition matrix (`mc_eg2`) converge to a stationary distribution — the Perron projection of the two matrices. + ```{code-cell} ipython3 P = np.array([[0.68, 0.12, 0.20], From 5bcfbf260e179bc0814ab46c0f9a2dfe47ce1e7a Mon Sep 17 00:00:00 2001 From: Humphrey Yang Date: Mon, 21 Apr 2025 17:40:59 +1000 Subject: [PATCH 2/3] minor updates --- lectures/eigen_II.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/lectures/eigen_II.md b/lectures/eigen_II.md index d78dca15..f406f715 100644 --- a/lectures/eigen_II.md +++ b/lectures/eigen_II.md @@ -238,7 +238,7 @@ A = \begin{bmatrix} 0.5 & 0.1 \\ \end{bmatrix} $$ -A here is also a primitive matrix since $A^k$ is everywhere positive for some $k \in \mathbb{N}$. +$A$ here is also a primitive matrix since $A^k$ is everywhere positive for some $k \in \mathbb{N}$. $$ @@ -393,7 +393,7 @@ We are now prepared to bridge the languages spoken in the two lectures. A primitive matrix is both irreducible and aperiodic. -So the Perron-Frobenius theorem explains why both the Imam and Temple matrix and Hamilton’s transition matrix (`mc_eg2`) converge to a stationary distribution — the Perron projection of the two matrices. +So Perron-Frobenius theorem explains why both {ref}`Imam and Temple matrix ` and {ref}`Hamilton matrix ` converge to a stationary distribution, which is the Perron projection of the two matrices ```{code-cell} ipython3 From b071ffe041a9425d098cfd4e11ec1d6407d5b298 Mon Sep 17 00:00:00 2001 From: Humphrey Yang Date: Mon, 21 Apr 2025 17:42:06 +1000 Subject: [PATCH 3/3] remove spaces --- lectures/eigen_II.md | 2 -- 1 file changed, 2 deletions(-) diff --git a/lectures/eigen_II.md b/lectures/eigen_II.md index f406f715..ce1c38b1 100644 --- a/lectures/eigen_II.md +++ b/lectures/eigen_II.md @@ -240,7 +240,6 @@ $$ $A$ here is also a primitive matrix since $A^k$ is everywhere positive for some $k \in \mathbb{N}$. - $$ B = \begin{bmatrix} 0 & 1 \\ 1 & 0 @@ -395,7 +394,6 @@ A primitive matrix is both irreducible and aperiodic. So Perron-Frobenius theorem explains why both {ref}`Imam and Temple matrix ` and {ref}`Hamilton matrix ` 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], [0.50, 0.24, 0.26],