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Math Help - Eigenvalue problems

  1. #1
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    Eigenvalue problems

    Conder the matrix
    A=[cos B -sin Q
    sin B cos B ]


    Multiplying a vector x in R^2 by A has the effect of rotationg x by angle B counter-clockwise about the origin.

    1. Reasoning geometrically, give two values of B in [ 0,2pi) for which A has real eigenvalues. For each such B, state the eigenvalues and corresponding eigenvalues associated with them.

    2.There is a value of B in [0,pi] for which A has i as an eigenvector. Find this value of B and find an eigenvector corresponding to the eigenvalue i in this case.

    3. The product of the eigenvalues of a matrix are always equal to the determinant of the matrix. Use this fact to determine the other eigenvalue of A for the value of B u find in part 2. Find a corresponding eigenvector.
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  2. #2
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    Quote Originally Posted by shannon1111 View Post
    Conder the matrix
    A=[cos B -sin Q
    sin B cos B ]


    Multiplying a vector x in R^2 by A has the effect of rotationg x by angle B counter-clockwise about the origin.

    1. Reasoning geometrically, give two values of B in [ 0,2pi) for which A has real eigenvalues. For each such B, state the eigenvalues and corresponding eigenvalues associated with them.
    A rotation keeps lengths the same so if Av= \lambda v, there are only two possible values of \lambda. And then it should be clear what the two angles are. (If one vector is a multiple of the other they are parallel. v must be rotated into a parallel vector.)

    2.There is a value of B in [0,pi] for which A has i as an eigenvector. Find this value of B and find an eigenvector corresponding to the eigenvalue i in this case.
    To get i as an eigenvalue (NOT eigenvector) we must be thinking of (x,y) as representing the complex number x+iy. Av= iv becomes A(x+iy)= i(x+iy). What is i(x+iy)? What point does that correspond to? What angle is (x,y) rotated through?

    3. The product of the eigenvalues of a matrix are always equal to the determinant of the matrix. Use this fact to determine the other eigenvalue of A for the value of B u find in part 2. Find a corresponding eigenvector.
    All rotations have the same determinant. What is it? (Hint: the identity transformation:x-> x is a rotation through 0 degrees.)
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