Non-diagonal matrix with eigenvalues equal to its diagonal entries

Question

Is it possible to construct a matrix with non-zero off-diagonal entries whose eigenvalues are nonetheless equal to its diagonal entries?

EDIT: @ajotataxe pointed out that this holds for triangular matrices. My follow up is - is the converse true? If the Eigen values are the diagonals, does it have to be triangular?

Answer 1

Yes:$$\begin{pmatrix}0&\frac12&\frac12\\1&1&1\\1&-1&-1\end{pmatrix}.$$Its eigenvalues are $0$, $1$ and $-1$.