Von Neumann Entropy

The Von Neumann Entropy is a generalisation of Shannon Entropy, useful for analysing the amount of information present in a Qubit. It is given as a function of the Density Operator $\rho$, namely $H(\rho )=-tr(\rho \ln \rho )$.

The value of $\ln \rho$ is given by the generalisation of the $\ln$ operator to matrixes. One way of doing this is to diagonalise the Density Operator, take the logarithms along the diagonal, and change the basis back to the original one.

I need to explain why we can diagonalise the matrix, or find a better explanation of $\ln \rho$. Help me ❓.

For a Density Operator $\rho$ equivalent to a pure Quantum state, the Von Neumann Entropy matches the Shannon Entropy.