Density operators are diagonalizable

We can write any Density Operator $\hat{\rho }=\sum {\rho }_i|{\psi }_i⟩⟨{\psi }_i|$, for a basis $\{|{\psi }_i⟩{\}}_i$.

Proof

Since the Density Operator $\hat{\rho }$ is an Hermitian Operator, and Hermitian operators are diagonalizable, then so is the Density Operator.