Condition for $(I+\epsilon A)(I+\epsilon A^\top)>\epsilon^2 AA^\top$

Question

Suppose that $A\in\mathbb{R}^{n\times n}$ is a square matrix. What is the condition for $$(I+\epsilon A)(I+\epsilon A^\top)>\epsilon^2 AA^\top.$$ Here, for matrices $X_1$ and $X_2$, $X_1>X_2$ means that $X_1-X_2$ is positive definite.

Any help would be highly appreciated!

Answer 1

Hint: if $B$ is symmetric, then the matrix $I+ \epsilon B$ is positive definite for sufficiently small $\epsilon.$