update solution and quote directives

Author: HumphreyYang

lectures/greek_square.md

@@ -4,7 +4,7 @@ jupytext:
     extension: .md
     format_name: myst
     format_version: 0.13
-    jupytext_version: 1.16.2
+    jupytext_version: 1.16.1
 kernelspec:
   display_name: Python 3 (ipykernel)
   language: python
@@ -16,12 +16,10 @@ kernelspec:
 
 ## Introduction
 
-
-
 Chapter 24 of {cite}`russell2004history` about early Greek mathematics and astronomy contains this
 fascinating passage:
 
- ```{note} 
+ ```{epigraph} 
  The square root of 2, which was the first irrational to be discovered, was known to the early Pythagoreans, and ingenious methods of approximating to its value were discovered. The best was as follows: Form two columns of numbers, which we will call the $a$'s and the $b$'s; each starts with a $1$. The next $a$, at each stage, is formed by adding the last $a$ and the $b$ already obtained; the next $b$ is formed by adding twice the previous $a$ to the previous $b$. The first 6 pairs so obtained are $(1,1), (2,3), (5,7), (12,17), (29,41), (70,99)$. In each pair, $2 a - b$ is $1$ or $-1$. Thus $b/a$ is nearly the square root of two, and at each fresh step it gets nearer. For instance, the reader may satisy himself that the square of $99/70$ is very nearly equal to $2$.
  ```
 
@@ -727,6 +725,7 @@ We shall encounter equations very similar to {eq}`eq:deactivate1` and {eq}`eq:de
 in {doc}`money_inflation` and in many other places in dynamic economic theory.
 
 
+## Exercise
 
 ```{exercise-start} 
 :label: greek_square_ex_a
@@ -745,7 +744,56 @@ compute the matrix $A$.
 ```{solution-start} greek_square_ex_a
 :class: dropdown
 ```
-Here is the answer
+Here is one soluition.
+
+According to the quote, we can formulate 
+
+$$
+\begin{aligned}
+a_{t+1} &= a_t + b_t \\
+b_{t+1} &= 2a_t + b_t
+\end{aligned}
+$$ (eq:gs_ex1system)
+
+with $x_0 = \begin{bmatrix}  a_0 \cr b_0 \end{bmatrix} = \begin{bmatrix}  1 \cr 1 \end{bmatrix}$
+
+By {eq}`eq:gs_ex1system`, we can write matix $A$ as 
+
+$$
+A = \begin{bmatrix} 1 & 1 \cr 
+                2 & 1 \end{bmatrix}
+$$
+
+Then $x_{t+1} = A x_t$ with $t \in \{0, \dots, 5\}$
+
+```{code-cell} ipython3
+# Define the matrix A
+A = np.array([[1, 1],
+              [2, 1]])
+
+# Initial vector x_0
+x_0 = np.array([1, 1])
+
+# Number of iterations
+n = 6
+
+# Generate the sequence
+xs = np.array([x_0])
+x_t = x_0
+for _ in range(1, n):
+    x_t = A @ x_t
+    xs = np.vstack([xs, x_t])
+
+# Print the sequence
+for i, (a_t, b_t) in enumerate(xs):
+    print(f"Iter {i}: a_t = {a_t}, b_t = {b_t}")
+
+# Compute eigenvalues and eigenvectors of A
+eigenvalues, eigenvectors = np.linalg.eig(A)
+
+print(f'\nEigenvalues:\n{eigenvalues}')
+print(f'\nEigenvectors:\n{eigenvectors}')
+```
 
 ```{solution-end}
-```
+```