Eigenvectors and eigevalues lecture ex needs improving

[jstac]

Regarding the first exercise in https://intro.quantecon.org/eigen_I.html, we need to make clear exactly what the reader should try to do. Steps:

1. Clarify the algorithm.
2. Give the reader a specific matrix and ask them to verify that the spectral radius is approximately x using the algorithm.

(That way they get the satisfaction of computing something before they check the solution.)

[longye-tian]

Dear John @jstac ,

Here is a version of the exercise I updated:

---

Power iteration is an algorithm used to compute the dominant eigenvalue of a diagonalizable $n \times n$ matrix $A$.

The method proceeds as follows:

1. Choose an initial random $n \times 1$ vector $b_0$.
2. Multiply $A$ by $b_k$ to obtain $Ab_k$.
3. Approximate the eigenvalue as $\lambda_k = \max|Ab_k|$.
4. Normalize the result: $b_{k+1} = \frac{Ab_k}{\max|Ab_k|}$.
5. Repeat steps 2-4 for $m$ iterations and $\lambda_k$, $b_k$ will converge to the dominant eigenvalue and its eigenvectors of $A$.

A thorough discussion of the method can be found here.

Implement the power iteration method using the following matrix and initial vector:

$$ A = \begin{bmatrix} 1 & 0 & 3 \\ 0 & 2 & 0 \\ 3 & 0 & 1 \end{bmatrix}, \quad \mathbf{b}_0 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} $$

Set the number of iterations to $m = 20$ and compute the dominant eigenvalue of $A$.

---

The solution for this exercise to match with the question

def power_iteration(A=np.array([[1, 0, 3],
              [0, 2, 0],
              [3, 0, 1]]), b=np.array([1,1,1]), m=20):
    
    for i in range(m):
        b = np.dot(A, b)
        lambda_1 = abs(b).max()
        b = b/ b.max()
    
    print('Eigenvalue:', lambda_1)
    print('Eigenvector:', b)

power_iteration()

What do you think about this update? Would you like to change anything?

Best ❤️
Longye

[jstac]

Hi @LongYe, this is great, many thanks!

I suggest replacinng "Multiply..." with "At the $k$-th iteration, multiply..."

Just to comply with the style manual, please use $b_0$ rather than $\mathbf b_0$.

[longye-tian]

Hi @LongYe, this is great, many thanks!

I suggest replacinng "Multiply..." with "At the k -th iteration, multiply..."

Just to comply with the style manual, please use b 0 rather than b 0 .

Sure thing. I will submit a PR for this issue and incorporate these suggestions. Thanks John!

Best ❤️
Longye