2025 OMOUS Problem 6

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2025 OMOUS Problem 6

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Source: Open Mathematical Olympiad for University Students (OMOUS-2025)

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#1 h Apr 18, 2025, 11:52 AM • 1 Y

Y by MS_asdfgzxcvb

Let $A=\left(a_{i j}\right)_{i, j=1}^{n} \in M_{n}(\mathbb{R})$ be a positive semi-definite matrix. Prove that the matrix $B=\left(b_{i j}\right)_{i, j=1}^{n} \text {, where }$ $b_{i j}=\arcsin \left(x^{i+j}\right) \cdot a_{i j}$ , is also positive semi-definite for all $x \in(0,1)$ .

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#2 h Apr 18, 2025, 10:43 PM

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By Schur's product theorem, we just have to show that $C=($ arcsin $(x^{i+j}))_{i,j}$ is semi-definite. In fact it's not hard to show that it actually works for any analytic function $f$ with positive Maclaurin coefficients.

This post has been edited 1 time. Last edited by Doru2718, Apr 24, 2025, 11:04 PM
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#3 h Apr 20, 2025, 9:06 PM

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@Doru2718 , How do you prove the theorem you quote above concerning the matrix $C=[f(x^{i+j})]_{i,j}$ ?

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#4 h Apr 24, 2025, 11:04 PM • 1 Y

Y by loup blanc

loup blanc wrote:

@Doru2718 , How do you prove the theorem you quote above concerning the matrix $C=[f(x^{i+j})]_{i,j}$ ?

If $M(f)$ denotes the matrix, then for $f = \sum_n a_n x^n$ , where $a_n>0$ we have $M(f) = \sum_n a_n M(x\to x^n)$ . So we only need to show that $M(x\to x^n) = (x^{n(i+j)})_{i,j}$ is semidefinite positive, for $x\in(0,1)$ . But this is just a trivial computation.

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#5 h Apr 27, 2025, 11:32 AM

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@Doru2718 , thanks; indeed, it's straightforward.

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