For a symmetric matrix $A$, I am aware that eigenvectors $v_1, \dots, v_n$ with the same eigenvalue $\lambda$ are linearly independent but not orthogonal.
The spectral theorem states that any $p \times p$ symmetric matrix has $p$ orthonormal eigenvectors. I do not understand how both these statements are correct when an eigenvalue can correspond to multiple eigenvectors and thus these eigenvectors can only be linearly independent with one another, and not orthogonal?