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

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!

Hint: if $$B$$ is symmetric, then the matrix $$I+ \epsilon B$$ is positive definite for sufficiently small $$\epsilon.$$
• But the point is that $A$ is not always symmetric and I want the requirement on $\epsilon$ and the singular values of $A$. Jun 28 at 3:18
• @lazyleo You are right! But $AA^T$ is symmetric. Jun 28 at 3:21
• @VezenBU I think you mean $A + A^T$. Jun 28 at 3:37