Conditions that a complex matrix has real and identical eigenvalues Given a matrix $H \in \mathbb{C}^{N \times N}$, is there some condition on its elements such that all the $N$ eigenvalues are real and have the same value? Obviously, the trivial case of the identity matrix is not acceptable. 
Thanks in advance.

Elaborating on Hagen von Eitzen's comment:
If an $N\times N$ matrix has $N$ eigenvalues equal to $\lambda$, we usually speak of a single eigenvalue $\lambda$ with geometric multiplicity $N$. The latter means that there exist $N$ linearly independent eigenvectors with eigenvalue $\lambda$. Since the space if $N$-dimensional, these eigenvectors already span the whole space. Now, vectors with eigenvalue $\lambda$ always form a subspace. All this means that the the whole space is the eigenspace corresponding to $\lambda$.
By definition, the matrix $A$ acts as $Ax = \lambda x$ on an eigenvector $x$. In our case, any vector is an eigenvector, hence $\forall x : Ax = \lambda x$, which is the same as $A = \lambda I$.
