The Gini index is well-known for a single measure of inequality. The purpose of this article is to explore a matrix structure of the Gini index in a setting of multiple source income. Using matrices, we analyze the decomposition of the Gini index by income source and derive an explicit formula for the factors in terms of the associated percentile levels based on aggregated data reporting. Each factor is shown to be the sums of the two split off parts of the income within a percentile bracket. Both have unequalizing and equalizing contribution to the total inequality, respectively. We use R code and apply the methodology to several data sets including a sample of European aggregated income reporting in 2014 for illustration. A byproduct from the Gini decomposition provides a matrix approach to the decomposition of the associated Lorenz curve in terms of the density distribution matrix and a Toeplitz matrix.

中文翻译：

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基尼系数按收入来源分解的汇总数据及其应用。

基尼系数因不平等的单一度量而闻名。本文的目的是在多种来源收入的情况下探索基尼指数的矩阵结构。使用矩阵，我们分析了按收入来源划分的基尼系数，并基于汇总数据报告，根据相关的百分位数水平得出了明确的因子公式。每个因素都显示为百分位数括号内收入的两个分割部分的总和。两者对总不平等的贡献分别为不均等和均等。我们使用R代码并将该方法应用于多个数据集，包括2014年欧洲汇总收入报告的示例，以进行说明。