Thread: 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!

Originally Posted by bamby

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 is an eigenvalue. The dimension of its eigenspace is determined by considering the nullspace of where is a matrix of ones. Clearly, the nullspace is . Also notice that is an eigenvalue since . But we have altogether counted at least eigenvalues with multiplicity. Thus, we found all of them.