How to square a density matrix?

Question

Supposing $x$ is a density matrix. I know purity formula is $$P = \mathrm{tr}(x^2)$$ But I have doubt about calculating $x^2$.

Is it $x\cdot x$ or $x\cdot x^\dagger$?

Can you give me a reference for your answer?

Answer 1

Squaring matrices works the same way as squaring numbers, i.e. you multiply the matrix by itself. Formally, $A^2=A\cdot A$. However, every density matrix $\rho$ is Hermitian, i.e. $\rho=\rho^\dagger$, so $\rho^2 = \rho\cdot\rho = \rho\cdot\rho^\dagger$, anyway.