# Solution: 2015-24 Hölder inequality for matrices

Let $$A, B$$ are $$n \times n$$ Hermitian matrices and $$p, q \in [1, \infty]$$ with $$\frac{1}{p} + \frac{1}{q} = 1$$. Prove that
$| Tr (AB) | \leq \| A \|_{S^p} \| B \|_{S^q}.$
(Here, $$\| A \|_{S^p}$$ is the $$p$$-Schatten norm of $$A$$, defined by
$\| A \|_{S^p} = \left( \sum_{i=1}^n |\lambda_i|^p \right)^{1/p},$
where $$\lambda_1, \lambda_2, \dots, \lambda_n$$ are the eigenvalues of $$A$$.)

The best solution was submitted by Park, Sunghyuk (박성혁, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-24.

Alternative solutions were submitted by 신준형 (2015학번, +3), 이정환 (2015학번, +3), 이종원 (수리과학과 2014학번, +3), 장기정 (수리과학과 2014학번, +3), 김동률 (2015학번, +2), 최인혁 (2015학번, +2).

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# Solution: 2015-23 Fixed points

Let $$f:[0,1)\to[0,1)$$  be a function such that $f(x)=\begin{cases} 2x,&\text{if }0\le 2x\lt 1,\\ 2x-1, & \text{if } 1\le 2x\lt 2.\end{cases}$ Find all $$x$$ such that $f(f(f(f(f(f(f(x)))))))=x.$

The best solution was submitted by Park, Sunghyuk (박성혁, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-23.

Alternative solutions were submitted by 김동률 (2015학번, +3), 신준형 (2015학번, +3), 유찬진 (2015학번, +3), 이상민 (수리과학과 2014학번, +3), 이종원 (수리과학과 2014학번, +3, his solution), 장기정 (수리과학과 2014학번, +3), 정호진 (동북고등학교 2학년, +3), 최인혁 (2015학번, +3), Daulet Kurmantayev (?, +3), 최동준 (포항공대 수학과 2013학번, +2).

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Let $$A, B$$ are $$n \times n$$ Hermitian matrices and $$p, q \in [1, \infty]$$ with $$\frac{1}{p} + \frac{1}{q} = 1$$. Prove that
$| Tr (AB) | \leq \| A \|_{S^p} \| B \|_{S^q}.$
(Here, $$\| A \|_{S^p}$$ is the $$p$$-Schatten norm of $$A$$, defined by
$\| A \|_{S^p} = \left( \sum_{i=1}^n |\lambda_i|^p \right)^{1/p},$
where $$\lambda_1, \lambda_2, \dots, \lambda_n$$ are the eigenvalues of $$A$$.)