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Suppose we are given a Hermitian matrix $A$, how to describe the following set of Hermitian $S=\{X:X\geq \pm A\}$, where $Y\geq B$ is $Y-B$ is semidefinite matrix.

This is of course a convex set, and my question is how to describe its boundary? We know that the boundary is not $|A|=\sqrt{A^+A}$ generally.

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  • $\begingroup$ Maybe, "$X\pm A$ has nontrivial kernel"? $\endgroup$
    – Alex Degtyarev
    Commented Feb 21, 2014 at 16:43
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    $\begingroup$ $\pm A \le X \leftrightarrow [X, A; A, X] \ge 0$ for Hermitian matrices $A$ and $X$. $\endgroup$
    – Suvrit
    Commented Jun 22, 2014 at 1:40

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Your set is simply described by the SDP $S=\{X: {\cal A}(x) \succeq {\cal B} \}$, where ${\cal A}(x) = \left[ \begin{array}{cc} X & 0 \\0& X \end{array} \right] $, and ${\cal B}=\left[ \begin{array}{c} A\\-A \end{array} \right]$. From this, the boundary is obtained by intersecting $S$ with the equation $\det({\cal A}(x)-{\cal B})=0$. This is not a very efficient description, but I think is the best you can do.

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