update graphviz and add more content to eigen2

Author: HumphreyYang

lectures/eigen_II.md

@@ -11,7 +11,7 @@ kernelspec:
   name: python3
 ---
 
-# Eigenvalues and Eigenvectors of Nonnegative matrices
+# Theorems of Nonnegative Matrices and Eigenvalues
 
 ```{index} single: Eigenvalues and Eigenvectors
 ```
@@ -20,39 +20,34 @@ kernelspec:
 :depth: 2
 ```
 
+In this lecture we will begin with the basic properties of nonnegative matrices.
+
+Then we will explore the Perron-Frobenius Theorem and the Neumann Series Lemma, and connect them to applications in Markov chains and networks. 
+
+We will use the following imports:
+
 ```{code-cell} ipython3
 import matplotlib.pyplot as plt
 import numpy as np
 from numpy.linalg import eig
 ```
 
+## Nonnegative Matrices
+
 Often, in economics, the matrix that we are dealing with is nonnegative.
 
 Nonnegative matrices have several special and useful properties.
 
 In this section we discuss some of them --- in particular, the connection
 between nonnegativity and eigenvalues.
 
-
-## Nonnegative Matrices
-
 Let $a^{k}_{ij}$ be element $(i,j)$ of $A^k$.
 
 An $n \times m$ matrix $A$ is called **nonnegative** if every element of $A$
 is nonnegative, i.e., $a_{ij} \geq 0$ for every $i,j$.
 
 We denote this as $A \geq 0$.
 
-### Primitive Matrices
-
-Let $A$ be a square nonnegative matrix and let $A^k$ be the $k^{th}$ power of A.
-
-A matrix is consisdered **primitive** if there exists a $k \in \mathbb{N}$ such that $A^k$ is everywhere positive.
-
-It means that $A$ is called primitive if there is an integer $k \geq 0$ such that $a^{k}_{ij} > 0$ for *all* $(i,j)$.
-
-This concept is closely related to irreducible matrices.
-
 ### Irreducible Matrices
 
 We have (informally) introduced irreducible matrices in the Markov chain lecture (TODO: link to Markov chain lecture).
@@ -61,8 +56,6 @@ Here we will introduce this concept formally.
 
 $A$ is called **irreducible** if for *each* $(i,j)$ there is an integer $k \geq 0$ such that $a^{k}_{ij} > 0$.
 
-We can see that if a matrix is primitive, then it implies the matrix is irreducible.
-
 A matrix $A$ that is not irreducible is called reducible.
 
 Here are some examples to illustrate this further.
@@ -74,9 +67,68 @@ Here are some examples to illustrate this further.
 3. $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ is reducible since $A^k = A$ for all $k \geq 0$ and thus
    $a^{k}_{12},a^{k}_{21} = 0$ for all $k \geq 0$.
 
-### Left and Right Eigenvectors
+### Primitive Matrices
+
+Let $A$ be a square nonnegative matrix and let $A^k$ be the $k^{th}$ power of $A$.
+
+A matrix is consisdered **primitive** if there exists a $k \in \mathbb{N}$ such that $A^k$ is everywhere positive.
+
+It means that $A$ is called primitive if there is an integer $k \geq 0$ such that $a^{k}_{ij} > 0$ for *all* $(i,j)$.
+
+We can see that if a matrix is primitive, then it implies the matrix is irreducible.
+
+This is becuase if there exists an $A^k$ such that $a^{k}_{ij} > 0$ for all $(i,j)$, then it guarantees the same property for ${k+1}^th, {k+2}^th ... {k+n}^th$ iterations.
+
+In other words, a primitive matrix is both irreducible and aperiodical as aperiodicity requires the a state to be visited with a guarantee of returning to itself after certain amount of iterations.
+
+### Left Eigenvectors
+
+We have previously discussed right (ordinary) eigenvectors $Av = \lambda v$.
+
+Here we introduce left eigenvectors.
+
+Left eigenvectors will play important roles in what follows, including that of stochastic steady states for dynamic models under a Markov assumption.
 
+We will talk more about this later, but for now, let's define left eigenvectors.
 
+A vector $\varepsilon$ is called a left eigenvector of $A$ if $\varepsilon$ is an eigenvector of $A^T$.
+
+In other words, if $\varepsilon$ is a left eigenvector of matrix A, then $A^T \varepsilon = \lambda \varepsilon$, where $\lambda$ is the eigenvalue associated with the left eigenvector $v$.
+
+This hints on how to compute left eigenvectors
+
+```{code-cell} ipython3
+# Define a sample matrix
+A = np.array([[3, 2], 
+              [1, 4]])
+
+# Compute right eigenvectors and eigenvalues
+right_eigenvalues, right_eigenvectors = np.linalg.eig(A)
+
+# Compute left eigenvectors and eigenvalues
+left_eigenvalues, left_eigenvectors = np.linalg.eig(A.T)
+
+# Transpose left eigenvectors for comparison (because they are returned as column vectors)
+left_eigenvectors = left_eigenvectors.T
+
+print("Matrix A:")
+print(A)
+print("\nRight Eigenvalues:")
+print(right_eigenvalues)
+print("\nRight Eigenvectors:")
+print(right_eigenvectors)
+print("\nLeft Eigenvalues:")
+print(left_eigenvalues)
+print("\nLeft Eigenvectors:")
+print(left_eigenvectors)
+left_eigenvectors @ right_eigenvectors
+```
+
+Note that the eigenvalues for both left and right eigenvectors are the same, but the eigenvectors themselves are different.
+
+We can then take transpose to obtain $A^T \varepsilon = \lambda \varepsilon$ and obtain $\varepsilon^T A= \lambda \varepsilon^T$.
+
+This is a more common expression and where the name left eigenvectors originates.
 
 ### The Perron-Frobenius Theorem
 
@@ -101,11 +153,18 @@ Moreover if $A$ is also irreducible then,
 4. the eigenvector $v$ associated with the eigenvalue $r(A)$ is strictly positive.
 5. there exists no other positive eigenvector $v$ (except scalar multiples of v) associated with $r(A)$.
 
+If $A$ is primitive then,
+6. the inequality $|\lambda| \leq r(A)$ is strict for all eigenvalues 𝜆 of 𝐴 distinct from 𝑟(𝐴), and
+7. with $e$ and $\varepsilon$ normalized so that the inner product of $\varepsilon$ and  $e = 1$, we have
+$ r(A)^{-m} A^m$ converges to $\varepsilon^{\top}$ when $m \rightarrow \infty$
 ```
 
 (This is a relatively simple version of the theorem --- for more details see
 [here](https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem)).
 
+Let's build our intuition for the theorem using a simple example.
+
+
 In fact, we have already seen Perron-Frobenius theorem in action before in the exercise (TODO: link to Markov chain exercise)
 
 In the exercise, we stated that the convegence rate is determined by the spectral gap, the difference between the largest and the second largest eigenvalue.

lectures/markov_chains_I.md

@@ -21,6 +21,7 @@ In addition to what's in Anaconda, this lecture will need the following librarie
 :tags: [hide-output]
 
 !pip install quantecon
+!pip install graphviz
 ```
 
 +++ {"user_expressions": []}

lectures/markov_chains_II.md

@@ -21,6 +21,7 @@ In addition to what's in Anaconda, this lecture will need the following librarie
 :tags: [hide-output]
 
 !pip install quantecon
+!pip install graphviz
 ```
 
 +++ {"user_expressions": []}