Matrix inequality $a X \succeq arcsin(X)$ for some $a > 0$

Let $$X \in S^{n}_{+}$$ be a positive semi-definite matrix with $$X_{ii} = 1$$ for all $$i \leq n$$ (thus $$X$$ is a correlation matrix). Since $$X$$ is positive semi-definite, we have $$|X_{ij}| \leq 1$$ for any entry $$(i,j)$$.

Let $$\arcsin(X) = (\arcsin(X_{ij}))_{i,j \leq n}$$, i.e., the arcsin function applies on the matrix X element-wisely. From the Schur product theorem, we know $$\arcsin(X) \succeq X$$, where $$A \succeq B$$ means $$A-B$$ is positive semi-definite. I wonder, do we have an inequality on the other side, i.e., there exists some real number $$a> 0$$ such that $$a X \succeq \arcsin(X)$$?

I am asking this question because SDP relaxation usually provides good approximation algorithms to combinatorial optimization with a maximization objective. I am thinking about if we can still apply SDP to minimization problems.

• 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 12 '21 at 0:52
• @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 '21 at 3:52