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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
    Eater of Worlds
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    Chaneysville, PA
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    For the first part, the Cayley-Hamilton theorem states that a square matrix satisifies its characteristic equation.

    The charpoly is {\lambda}^{2}-5{\lambda}+8=0

    A=\begin{bmatrix}3&-2\\1&2\end{bmatrix}

    A^{2}=\begin{bmatrix}7&-10\\5&2\end{bmatrix}

    So, confirm that A^{2}-5A+8=0
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