An exercise question (9.7) from Quantum computation and Quantum Information by Michael E. Nielson and Isaac L. Chuang says that I can write the difference of any 2 arbitrary density operators $\rho,\sigma$ as a spectral decomposition: $$ \rho - \sigma = U D U^\dagger $$ But for this to be true, isn't it necessary that the density matrices must commute? From what I understand, I will need an orthonormal basis in which both will be diagonalizable. Based on the Simultaneous diagonalizable theorem this is only the case when $\rho$ and $\sigma$ commute.
What am I missing?