Day 40: Linear Algebra

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Announcements:
- Corrected exams are reviewed. A key is available. Please check it out!
  - I want to examine Emily's proof, #6.
- Sorry: Homework isn't graded yet.
- Your second project is due next time, Friday.
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?

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