Timeline for Matrix inequality $a X \succeq arcsin(X)$ for some $a > 0$
Current License: CC BY-SA 4.0
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| Feb 12, 2021 at 3:52 | comment | added | zxzx179 | @fedja I see. I think it's a great example. Here $arcsin(X)$ is positive definite but $X$ is only positive semi-definite. We can find an invertible matrix $P$ such that $arcsin(X) = P^{T} P$ and $X = P^{T} D P$, with $D$ being a diagonal matrix with some diagonal entry being zero. Thus, $aX - arcsin(X) = P^{T} (aD - I) P$ can not be positive semi-definite. | |
| Feb 12, 2021 at 0:52 | comment | added | fedja | A trivial $3\times 3$ counterexample is $\begin{bmatrix}1&-0.5&-0.5\\-0.5&1&-0.5\\-0.5&-0.5&1\end{bmatrix}$. $1\times 1$ is certainly fine and $2\times 2$ seems OK too though I haven't checked it thoroughly. | |
| Feb 11, 2021 at 23:03 | history | edited | YCor | CC BY-SA 4.0 |
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| Feb 11, 2021 at 22:23 | history | edited | zxzx179 | CC BY-SA 4.0 |
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| Feb 11, 2021 at 22:18 | history | asked | zxzx179 | CC BY-SA 4.0 |