关于两个几何平均与伴随轨道之和
On two geometric means and sum of adjoint orbits
摘要：
本文研究了由Pusz和Woronowicz引入的度量几何平均和由Fiedler和Pták引入的谱几何平均，它们最初是针对正定矩阵的.通过对数优化建立了$t$ -度量几何平均和$t$ -谱几何平均之间的关系。然后将结果推广到与非紧半单李群相关的对称空间中。对于任何厄米矩阵$X$和$Y$，So的矩阵指数公式断言存在酉矩阵$U$和$V$，使得$$e^{X/2}e^Ye^{X/2} = e^{UXU^*+VYV^*}.$$换句话说，厄米矩阵$\log (e^{X/2}e^Ye^{X/2})$位于$X$和$Y$的酉轨道之和。So的结果也被推广到非紧半单李群的伴随轨道的一个公式。
英文摘要：
In this paper, we study the metric geometric mean introduced by Pusz and Woronowicz and the spectral geometric mean introduced by Fiedler and Pták, originally for positive definite matrices. The relation between $t$-metric geometric mean and $t$-spectral geometric mean is established via log majorization. The result is then extended in the context of symmetric space associated with a noncompact semisimple Lie group. For any Hermitian matrices $X$ and $Y$, So's matrix exponential formula asserts that there are unitary matrices $U$ and $V$ such that $$e^{X/2}e^Ye^{X/2} = e^{UXU^*+VYV^*}.$$ In other words, the Hermitian matrix $\log (e^{X/2}e^Ye^{X/2})$ lies in the sum of the unitary orbits of $X$ and $Y$. So's result is also extended to a formula for adjoint orbits associated with a noncompact semisimple Lie group.
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