Day 43: Linear Algebra

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Today:

Announcements:
- It is time to evaluate the course. I'm sure that you've received emails to that effect....
- Your second project is returned. There were some very nice ones! What do you think of this particular application of linear algebra?
- Your determinant assignment is returned:
  - 1.6, p. 296
  - 2.8, p. 299
    - Gauss's method seeks to reduce the matrix to diagonal form -- then determinants are easy: just the product of the diagonal elements.
    - On this one, you should realize that, by the time you get to a diagonal matrix, all you have to do is multiply all the diagonal elements (then take care of any swaps and scales).
  - 2.13 -- as soon as you realize that the matrix is singular, you're done. The author had a good idea -- it's better than simply computing the determinant!
- There is one final homework assignment, due Friday. We'll look over the homework in class.
- Next time we'll have a review of the course in preparation for the final.
Chapter Five: Similarity
- Our objective is to understand Eigenvalues and Eigenvectors
  - The chapter starts off with complex numbers. Why would they be necessary? We'll see in a bit....
  - Let's review our example. Consider the homomorphism taking into , given by the matrix defined by ${H_{E_2,E_2}=\left ( \begin{array}{cc} {13/3}&{ -2/3}\cr {-4/3}&{11/3} \end{array} \right ) }$
    - First, what happens when we multiply the two vectors ${\langle \left(\begin{array}{c} {1}\cr {2} \end{array}\right),\left(\begin{array}{c} {-1}\cr {1} \end{array}\right) \rangle}$ by H?
    - What's a better basis for both the domain and codomain in which to represent this transformation?
    - Okay, so we've found some vectors that have the property that their images under the transformation h are actually just scalar multiples of themselves.
      These vectors have a special name: eigenvectors. And the scalars by which they're scalled are called eigenvalues.
    - Now, an important question is this: under what conditions can we accomplish this this "diagonalization"? Under what conditions can we find eigenvectors and eigenvalues that allow us to construct this very simple matrix representation of the homomorphism?
      Are all matrices "diagonalizable"? (Can you think of a homomorphism from ${\R^2\rightarrow\R^2}$ we've studied geometrically where no vector will have as an image a multiple of itself?
    - The vectors I proposed for multiplication by the matrix H came "out of the blue". How would we go about finding them? Let's think about what we're looking for in terms of an equation: ${H\overline{v}=\lambda\overline{v}}$
      We can re-write that as
      
      ${H\overline{v}=\lambda{I}\overline{v}}$
      obviously (since I is the identity matrix); now we take everything over to the lefthand side,
      
      ${\left(H-\lambda{I}\right)\overline{v}=\overline{0}}$
      i.e.
      
      ${H_{E_2,E_2}=\left ( \begin{array}{cc} {13/3-\lambda}&{ -2/3}\cr {-4/3}&{11/3-\lambda} \end{array} \right )\left(\begin{array}{c} {x}\cr {y} \end{array}\right)=\left(\begin{array}{c} {0}\cr {0} \end{array}\right) }$
  - Now, let's consider another example: a diagonal matrix
    - What are its eigenvalues?
    - What are its eigenvectors?
    - Well, we know how to find them....
  - Finally, let's consider a more interesting example: a rotation matrix:
    
    ${H_{E_2,E_2}=\left ( \begin{array}{cc} {\cos(\theta)}&{-\sin(\theta)}\cr {\sin(\theta)}&{\cos(\theta)} \end{array} \right )}$
    - What are the eigenvalues and eigenvectors?
      - Last time we found the eigenvalues: ${\cos(\theta)\pm{i}\sin(\theta)}$
      - Now we'll find the eigenvectors corresponding to these eigenvalues.
    - A slight complex digression: Euler's formula
      You may have encountered Euler's Formula at some point in your life:
      
      ${e^{i\theta}=\cos(\theta)+i\sin(\theta)}$
- Last time we looked at a "practical" example: Topic: Stable Populations, p. 403
  - The transition matrix between park was given by the two-by-two matrix in the following system:
    
    ${\left [ \begin{array}{c} {p_{n+1}}\cr {r_{n+1}} \end{array} \right ] = \left [ \begin{array}{cc} {.9}&{.01}\cr {.1}&{.99} \end{array} \right ] \left [ \begin{array}{c} {p_{n}}\cr {r_{n}} \end{array} \right ] }$
  - We discovered that there is a population ratio that is stable under this transition, given by the first eigenvector.
  - We computed the eigenvectors by hand: 1 and .89.
  - The author uses the relatively unfamiliar (and non-standard) greek letter ${\overline{\zeta}}$ (zeta) for eigenvectors. The greek letter ${\lambda}$ is traditional for eigenvalues. Last time we found the eigenvectors, too: I chose ${\left [ \begin{array}{c} {1}\cr {10} \end{array} \right ] }$ and ${\left [ \begin{array}{c} {1}\cr {-1} \end{array} \right ] }$
    We can use these to "diagonalize" the matrix. Let's write that out, and see how it works. It helps us to look at the long term behavior of the system.
  - I want to quibble with the author a little, in that as growth in the world population occurs, the park population will chase this stable ratio of animals. The stable ratio depends only on the transition matrix, not on the absolute population of animals.
    So we can make this system more realistic as a model for population ecology. Be we don't focus on the absolute population vector.
- A second, more theoretical example: Topic: Fibonacci Numbers, p. 405
  - The historical background
  - Setting up the transition matrix
  - A formula for the calculation of the ${n^{th}}$ Fibonacci number.
- A final example: the linear regression problem
  - In the regression problem, the matrix we must diagonalize is symmetric. What can we say about the eigenvalues of a symmetric matrix?
  - We'll just consider the 2x2 case: ${\left [ \begin{array}{cc} {a}&{b}\cr {b}&{d} \end{array} \right ]}$
Links:
- Eigenvalues and Eigenvectors of a rotation matrix (this author uses clockwise rotations -- sigh....:)
- One man's favorite Eigenvalue problems

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