We say that a matrix $P$ with non-negative entries majorizes another such matrix $Q$ if there is a stochastic matrix $T$ such that $Q=TP$. We study matrix majorization in large samples and in the catalytic regime in the case where the columns of the matrices need not have equal support, as has been assumed in earlier works. We focus on two cases: either there are no support restrictions (except for requiring a non-empty intersection for the supports) or the final column dominates the others. Using real-algebraic methods, we identify sufficient and almost necessary conditions for majorization in large samples or when using catalytic states under these support conditions. These conditions are given in terms of multi-partite divergences that generalize the R\'enyi divergences. We notice that varying support conditions dramatically affect the relevant set of divergences. Our results find an application in the theory of catalytic state transformation in quantum thermodynamics.

翻译：当存在随机矩阵$T$使得$Q=TP$时，我们称具有非负元素的矩阵$P$主要化另一同类矩阵$Q$。本研究探讨大样本及催化体系下的矩阵主要化问题，其中矩阵的列无需具有相同支撑集——此条件在先前研究中曾被假定。我们聚焦于两种情形：要么不存在支撑集限制（仅要求支撑集存在非空交集），要么末列支配其他列。运用实代数方法，我们在这些支撑条件下，识别出了大样本或使用催化态时主要化的充分且近乎必要的条件。这些条件以多部散度的形式给出，其推广了Rényi散度。我们注意到，变化的支撑条件会显著影响相关散度集合的构成。我们的研究结果在量子热力学催化态转换理论中具有应用价值。