1. ## 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!

2. 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 $\displaystyle 0$ is an eigenvalue. The dimension of its eigenspace is determined by considering the nullspace of $\displaystyle A$ where $\displaystyle A$ is a matrix of ones. Clearly, the nullspace is $\displaystyle n-1$. Also notice that $\displaystyle n$ is an eigenvalue since $\displaystyle T(1,...,1) = n(1,...,1)$. But we have altogether counted at least $\displaystyle n$ eigenvalues with multiplicity. Thus, we found all of them.