[Math] Find the eigenvalues and eigenvectors with zeroes on the diagonal and ones everywhere else.

Question

I have been working on this problem for a couple hours and am completely stuck. Any help would be greatly appreciated.

Let $A$ be the $n \times n$ matrix which has zeros on the main diagonal and ones everywhere else. Find the eigenvalues and eigenvectors of $A$.

Answer 1

Note that if $\lambda=-1$, then $$ A-\lambda I = \begin{bmatrix} 1 & \dotsc & 1 \\ \vdots & \ddots & \vdots \\ 1 & \dotsc & 1 \end{bmatrix} $$ But $A-\lambda I$ has rank $1$, so $\lambda_1=-1$ is an eigenvalue of $A$ with geometric multiplicity $n-1$. The eigenspace $E_{\lambda_1}$ is the nullspace of this matrix. Can you find it? Hint: it has dimension $n-1$.

Now, note that $\operatorname{trace}A=0$ so $\lambda_2=n-1$ is the other eigenvalue of $A$. The eigenspace $E_{\lambda_2}$ is the nullspace of $A-\lambda_2 I$. Can you find it too?