I'm reading a paper [1], which provides a condition for checking whether the square of a matrix is positive semi-definite: suppose that $A \in \mathbb R^{n \times n}$ is a real-valued square matrix, then the symmetric part of $A^2 = A A$ is positive semi-definite as long as $A$ is positive semi-definite and $$ \|A - A^{T} \|_2 \leq \|A + A^{T} \|_2. $$
This proposition is given in a remark without proof. Appreciate it if anyone can help to prove.
[1] On the Convergence of Projection Methods: Application to the Decomposition of Affine Variational Inequalities