A note on the matrix arithmetic-geometric mean inequality
A note on the matrix arithmetic-geometric mean inequality
复制标题
关于矩阵算术几何平均不等式的注解
DOI：
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
Teng Zhang
中科院分区：
文献类型：
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作者：
Teng Zhang
This note proves the following inequality: If $n=3k$ for some positive integer $k$, then for any $n$ positive definite matrices $A_1,A_2,dots,A_n$, the following inequality holds: egin{equation*}label{eq:main} frac{1}{n^3} , Big|sum_{j_1,j_2,j_3=1}^{n}A_{j_1}A_{j_2}A_{j_3}Big| ,geq, frac{(n-3)!}{n!} , Big|sum_{substack{j_1,j_2,j_3=1,\ ext{$j_1$, $j_2$, $j_3$ all distinct}}}^{n}A_{j_1}A_{j_2}A_{j_3}Big|, end{equation*} where $|cdot|$ represents the operator norm. This inequality is a special case of a recent conjecture proposed by Recht and R'{e} (2012).
This note proves the following inequality: If $n=3k$ for some positive integer $k$, then for any $n$ positive definite matrices $A_1,A_2,dots,A_n$, the following inequality holds: egin{equation*}label{eq:main} frac{1}{n^3} , Big|sum_{j_1,j_2,j_3=1}^{n}A_{j_1}A_{j_2}A_{j_3}Big| ,geq, frac{(n-3)!}{n!} , Big|sum_{substack{j_1,j_2,j_3=1,\ ext{$j_1$, $j_2$, $j_3$ all distinct}}}^{n}A_{j_1}A_{j_2}A_{j_3}Big|, end{equation*} where $|cdot|$ represents the operator norm. This inequality is a special case of a recent conjecture proposed by Recht and R'{e} (2012).