# Finding eigenvalues and eigenstates

I am currently taking a course in Quantum Mechanics and I am getting confused by the start of this question, I'm sure it's fairly simple but I'm not getting it.

The question is:

A quantum mechanical system is described by a two dimensional Hilbert space of states, spanned by an orthonormal basis $|\alpha\rangle,|\beta\rangle$, with the following Hamiltonian:

$$H|\alpha\rangle = 4|\alpha\rangle + |\beta\rangle , H|\beta\rangle = |\alpha\rangle+4|\beta\rangle$$

I need to first find the eigenvalues and eigenstates of the hamiltonian. My intuition tells me this is possible without directly working out the matrix, but I would also like to know the matrix. I also know that $|\alpha\rangle$ and $|\beta\rangle$ are orthogonal, since a quantum hamiltonian is hermitian and admits and orthogonal basis.

The matrix is $\left[\begin{matrix}4 & 1\\1&4\end{matrix}\right]$