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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!

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1 Answer 1

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

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  • $\begingroup$ But the point is that $A$ is not always symmetric and I want the requirement on $\epsilon$ and the singular values of $A$. $\endgroup$
    – lazyleo
    Jun 28 at 3:18
  • $\begingroup$ @lazyleo You are right! But $AA^T$ is symmetric. $\endgroup$
    – Vezen BU
    Jun 28 at 3:21
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    $\begingroup$ @VezenBU I think you mean $A + A^T$. $\endgroup$ Jun 28 at 3:37
  • $\begingroup$ @lazyleo Expand the left-hand side, then think about how the hint can be utilised. $\endgroup$ Jun 28 at 6:34

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