We are interested in the distribution of Wishart samples after forgetting their scaling factors. We call such a distribution a projective Wishart distribution. We show that projective Wishart distributions have strong links with the affine-invariant geometry of symmetric positive definite matrices in the real case or Hermitian positive definite matrices in the complex case. First, the Fr{\'e}chet mean of a projective Wishart distribution is the covariance parameter, up to a scaling factor, of the corresponding Wishart distribution. Second, in the case of 2 by 2 matrices, the densities have simple expressions in term of the affine-invariant distance.

翻译：我们关注Wishart样本在忽略其缩放因子后的分布特性，此类分布称为投影Wishart分布。研究表明，投影Wishart分布与实对称正定矩阵或复Hermitian正定矩阵的仿射不变几何具有深刻联系。首先，投影Wishart分布的Fr{\'e}chet均值即为对应Wishart分布的协方差参数（相差一个缩放因子）。其次，对于2×2矩阵情形，其概率密度函数可通过仿射不变距离简洁表达。