[eigen_I] Improvement suggestions

[HumphreyYang]

This PR collects feedback from the reading group on eigen_I.

Content

• [ ] Add an introduction to what eigenvalues and eigenvectors are. We can give the formula at the beginning and a visualization showing that eigenvectors are invariant in the transformation and eigenvalues are the scaling factors. We can link this further to the vector field plots in the exercises.
• [ ] Give a better motivation for studying eigenvalues and eigenvectors: (1) mention that it is used in Markov Chain and more advanced lectures in intermediate series; (2) add links to the list of areas that use eigenvalues below: "At the same time, these concepts are extremely useful for".
• [ ] Transformation plots (in https://intro.quantecon.org/eigen_I.html#square-matrices) should be hidden and potentially relocated, and we should explain the details of the transformation (@jstac).
• [ ] Hide functions used to generate the visualizations and reference the function name in the main text.
• [ ] Reformat the subtitles to meet the style sheet (only the first word and proper names should be capitalized).
• [ ] We can give an overview of what is the Neumann Series Lemma at the beginning of the section.

Code

• [ ] Some labels in the graphs are not in the math mode $ $.
• [ ] In this plot, the label for the transformed plot should be $Ax_1, Ax_2, ..., Ax_k$ instead of $Bx_1, Bx_2, ..., Bx_k$, and the second label should be $BAx_1, BAx_2, ..., BAx_k$
• [ ] We can remove the default values in

  def grid_composition_transform(A=np.array([[1, -1], [1, 1]]),
                             B=np.array([[1, -1], [1, 1]])):

  .

• [ ] Rewrite the code in exercise 7.1 to using while loop and error tolerance.

[jstac]

The vector field suggestion is really good. I think we could almost start off with a picture something like https://en.wikipedia.org/wiki/Vector_field#/media/File:VectorField.svg where the arrows show motion from $x$ to $Ax$.

Then we note that some arrows exhibit a scaling action. We can add in lines that show the eigenvectors (that are scaled) more clearly. Then we can say that the vectors on these lines are eigenvectors, and the scaling factors are the eigenvalues.

Then we can discuss how they are important, following @HumphreyYang 's point "Give a better motivation for ..."

[HumphreyYang]

Further comments by @pgrosser1:

• [ ] Around 8.4.1, it might be helpful to note that matrix multiplication can be commutative in specific cases (which happen to be extraordinarily important for physical systems). (@HumphreyYang: we can link to https://en.wikipedia.org/wiki/Commuting_matrices)
• [ ] At the end of 8.6.1, it would be helpful to also include a visual representation of what happens when a vector that is not an eigenvector of a matrix is multiplied by that matrix, by visualizing the original vector, the resultant vector, and the eigenvectors of the matrix on the same diagram (a common visualization utilizes a parallelogram representation of the matrix eigenvectors).