Day 41: Linear Algebra

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Announcements:
- Your second project is due today.
- Your homework is graded.
  - 1.8 -- why is 1.8a unusual? The inverse of this reflection is itself! What other obvious linear transformations have this same property?
  - 1.9
  - 1.17 -- but I threw it out, because it really requires an unstated assumption. As usual, lurking behind all the other bases is the standard basis.... (See 1.7/1.8d, for example of this lurking!)
- You have an assignment due Monday (this was pushed back).
- I've added one final homework assignment, which will be due Friday of next week. We'll look over the homework in class.
  It involves the calculations of some eigenvalues and eigenvectors.
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:
      
      ${e^{i\theta}=\cos(\theta)+i\sin(\theta)}$
- Now we'll look at a practical example: Topic: Stable Populations, p. 403
  - We'll discover that there is a population ratio that is stable under this transition.
  - Let's compute the eigenvectors by hand.
  - I'll 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.
  - The author uses the relatively unfamiliar (and non-standard) greek letter ${\overline{\zeta}}$ (zeta) for eigenvectors. The greek letter ${\lambda}$ is traditional for eigenvalues.
Links:
- Eigenvalues and Eigenvectors of a rotation matrix (this author uses clockwise rotations -- sigh....:)
- One man's favorite Eigenvalue problems

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