So, I was studying quantum computation using the book Nielsen and Chuang and it stated a theorem known as "Spectral Decomposition theorem"

$$A=\sum _{i}\lambda _{i} | i \rangle \langle i|$$

I infer from this theorem that any normal operator can be diagonalized in the basis set $ \left\{ |i \rangle \right\} $ which should be the eigen vectors of the operator matrix A with $ \lambda _{i}$ as the eigenvalues.

Now when I started studying about the density operator with the definition $$\rho = \sum _{i}p_{i} |\psi _{i} \rangle \langle \psi _{i}|$$ I got a little confused. Since $\rho$ is a normal operator and it can be written as this decomposition, it must mean that the vectors $| \psi _{i} \rangle$ must be orthonormal to each other according to the spectral decomposition theorem. This seems totally absurd to me since there is no reason for the pure states (combining to make a mixed state) to have orthonormality as a prerequisite. I am sorry if my question is very trivial as this is my first time studying quantum information and I would be glad if someone could help me with this problem.