So everyone knows eigenvectors corresponding to different eigenvalues are orthogonal to each other, given that the operator is self-adjoint.
If we have a self-adjoint operator, say $L$, is it possible that $\exists u, v$ such that $Lu=\lambda u$, $Lv=\lambda v$ and $\langle u, v\rangle=0$. In other words, we have eigenvectors with the same eigenvalues to $L$ and they are still orthogonal?
This is motivated by considering the angular momentum operators in Quantum Mechanics, I was wondering if there is a simpler example in Linear Algebra.
Best Answer
Just take the $L$ to be the identity operator on a finite dimensional Hilbert space (inner product space). And use the orthonormal basis basis .
If your eigen-space has dimension greater than $1$ then it is always possible. Just take the orthonormal basis of the eigen-space.