As a foreword, this is part of my homework so I have left all the calculations out: I just need an explanation.
I was trying to show that D = U†AU is diagonal, where D is the diagonalized matrix, A is a Hermitian matrix, U is found using the eigenvectors and U† is of course the conjugate transpose of U. I got D = [1,-3 ; -3,1] but my question is how do I know if this is diagonal or not? Is it diagonal because it reduces to [1,0 ; 0,1] and so the non diagonal entries are zero?
A diagonal matrix is an a square matrix $D = [d_{ij}]$ with $d_{ij} = 0$ for $i \neq j$; that is, a square matrix which can only have non-zero entries on the diagonal. The matrix $$\left[\begin{matrix}1 & -3\\ -3 & 1\end{matrix}\right]$$ is not diagonal. Your proposed definition of diagonal is equivalent to invertibility. You must have made a mistake in your evaluation of the product $U^{\dagger}AU$, or possibly in determining $U$ itself.