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Math Help - Help with Linear Algebra(Eigenvalues)

  1. #1
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    Help with Linear Algebra(Eigenvalues)

    I'm having trouble with a few of the harder linear algebra concepts from the eigenvalue chapter.
    1) Using the matrix A= | 3 -2 |
    | 1 2 |
    verify the Cayley Hamilton theorem.

    2) Given the rotation matrix Rtheta = | cos -sin|
    | sin cos|
    show the matrix has eigenvectors and eigenvalues corresponding to
    lambda= e^i*theta : |1|
    |-i|
    lambda= e^-i*theta |1|
    |i |
    3)Show that similar matrices A and B have the same eigenvalues. Thus you must show that det(A-lambda*I)= det(B-lambda*I)

    Thanks to anyone takes a look at this.
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  2. #2
    Senior Member Peritus's Avatar
    Joined
    Nov 2007
    Posts
    397
    I'm not saying google should become your best fried but at least make it one of your friends:

    Cayley–Hamilton theorem - Wikipedia, the free encyclopedia

    Eigenvalue, eigenvector and eigenspace - Wikipedia, the free encyclopedia

    3. if A and B are similar then there exists and invertible mtrix P such that:

    P^{ - 1} AP = B


    \begin{gathered}<br />
  \left| {B - \lambda I} \right| = \left| {P^{ - 1} AP - \lambda I} \right| = \left| {P^{ - 1} AP - P^{ - 1} \lambda IP} \right| = \left| {P^{ - 1} \left( {A - \lambda I} \right)P} \right| \hfill \\<br />
   = \left| {P^{ - 1} } \right|\left| {A - \lambda I} \right|\left| P \right| = \left| {A - \lambda I} \right| \hfill \\<br />
   \hfill \\<br />
  Q.E.D. \hfill \\ <br />
\end{gathered}
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