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Math Help - eigenvalues

  1. #1
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    Wink eigenvalues

    Hi,
    Would you help me wuth this, please!
    Suppose n is a positive integer and T is in F^n is defined by
    T(z_1, z_2, ... , z_n) = (z_1+ ... +z_n, z_1+ ... +z_n, ...............,z_1+ ... +z_n)
    Determine all eigenvalues and eigenvectors of T.
    Thank you!
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  2. #2
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    Quote Originally Posted by bamby View Post
    Hi,
    Would you help me wuth this, please!
    Suppose n is a positive integer and T is in F^n is defined by
    T(z_1, z_2, ... , z_n) = (z_1+ ... +z_n, z_1+ ... +z_n, ...............,z_1+ ... +z_n)
    Determine all eigenvalues and eigenvectors of T.
    Thank you!
    We note that 0 is an eigenvalue. The dimension of its eigenspace is determined by considering the nullspace of A where A is a matrix of ones. Clearly, the nullspace is n-1. Also notice that n is an eigenvalue since T(1,...,1) = n(1,...,1). But we have altogether counted at least n eigenvalues with multiplicity. Thus, we found all of them.
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