What are the eigenvectors of this matrix and if these are the eigenvectors, why aren't they orthogonal?
Given this matrix:
$$\begin{pmatrix} 0 & 1\\b & 0 \end{pmatrix}$$.
with $b \in \mathbb{R}$
I got the distinct eigenvalues to be $\pm \sqrt b$.
At the moment I get that they're:
$\begin{pmatrix} 1 \\ \sqrt b \end{pmatrix}$ and $\begin{pmatrix} 1 \\ - \sqrt b \end{pmatrix}$
But these aren't orthogonal?
Why not?
Because the matrix is not symmetric. A set of orthogonal eigenvectors that span the space is only guaranteed when the matrix is symmetric (or, in the complex case, when it is Hermitian.)