Eigenvalues and eigenvectors computation

Eigenvalues and eigenvectors computation

Let $T$ be the $n \times n$ matrix with every entry equal to $1$. I
computed the EV and EVs as follows and was wondering if it is correct:
Since every component of $Tv$ equals $v_1 + ... +v_n$ it is evident that
$n$ is an eigenvalue and $(1)$ is a corresponding eigenvector. Next I
observed that every column is a multiple of the first. Hence the null
space of $T$ has dimension $n-1$. As far as I understand the null space is
the eigenspace corresponding to the eigenvalue $0$ (if it is non-trivial).
Therefore to find eigenvectors corresponding to $0$ it is enough to find
$n-1$ linearly independent vectors. It is evident that the vectors $v_i$
of the form $1$ at the first component and $-1$ at $i$ for $i>1$ satisfy
this requirement. Thank you for checking my solution.