I am taking a quantum mechanics class and was assigned this problem:

Among other things, I am asked to find the eigenvalues of $H$ in terms of $a$, $b$ and $\sigma$. I'm sort of lost of even how to approach this.

Since $\hat{A}$ is Hermitian, I am assuming that $|a\rangle$ and $|b\rangle$ are orthonormal and complete. After that I'm pretty lost about even how to start. I have roughly written down that

$ \hat H | \lambda_n\rangle = \lambda_n|\lambda_n\rangle$ where $\lambda$ is an eigenvalue and $|\lambda_n\rangle$ is an eigenvector. What I don't grasp at all, is how to reconcile the outer product,

$|a\rangle \langle b| + |b\rangle \langle a| $ since that result is a matrix, right?