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$e^{X/2}{e}^Y{e}^{X/2}=e^{UX{U}^{⁎}+VY{V}^{⁎}}.$ In other words, the Hermitian matrix $\log (e^{X/2}{e}^Y{e}^{X/2})$ 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.

在本文中，我们研究了 Pusz 和 Woronowicz 引入的度量几何均值以及 Fiedler 和 Pták 引入的谱几何均值，最初用于正定矩阵。t -度量几何平均数和t -谱几何平均数之间的关系是通过对数专业化建立的。然后在与非紧半单李群相关的对称空间的上下文中扩展结果。对于任何 Hermitian 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${\mathrm{电子}}^{X/2}{\mathrm{电子}}^{是}{\mathrm{电子}}^{X/2}={\mathrm{电子}}^{你X{你}^{⁎}+伏是{伏}^{⁎}}.$ 换句话说，厄米矩阵 $\mathrm{日志}({\mathrm{电子}}^{X/2}{\mathrm{电子}}^{是}{\mathrm{电子}}^{X/2})$位于X和Y的酉轨道之和。So 的结果也扩展到与非紧半单李群相关的伴随轨道公式。