Essence of Linear Algebra: Publish Quick Eigen

[kurtbruns]

This pull request:

• Publishes the Quick Eigen article for review

[3b1b]

One bigger change I might recommend is to deviate from the video structure and put the trick and exercise up at the top of the article.

• Show the trick
  • Given a matrix [[a, b], [c, d]], even if you can't immediately see its eigenvalues, you can almost read off the mean and product of those eigenvalues:
    • Mean = mean(diagonal) = (a + d) / 2
    • Product = det = ad - bc
  • Given the mean and product of two numbers, you can recover them with m±sqrt(m^2 - p). [Acknowledge that this is essentially the quadratic formula]
• Give the multiple-choice questions
• Explain why the trick works
  • Review the usual way to compute them by subtracting lambda off diagonals, expanding the det, and solving the characteristic polynomial.
  • Facts about det, trace, etc.
• QM example

[3b1b]

Also, while reading this, I noted a slightly faster way. Let's write a general matrix as [[a, b], [c, d]], with the specific example [[2, 4], [6, 8]] to illustrate

1. Write a, d as m ± s. For example, 2, 8 = 5 ± 3
2. The eigenvalues are m ± sqrt(s^2 + bc). For example 5 ± sqrt(9 + 4 * 6) = 5 ± sqrt(33)

I'm hesitant to add much more because the trick doesn't really matter, it's the exercise in thinking about why it works that matters.