It is frequently mentioned in literature that LCA is linear, without a proof, or even without a clear definition of the criterion for linearity. Here we study the meaning of the term linear, and in relation to that, the question if LCA is indeed linear. We explore the different meanings of the term linearity in the context of mathematical models. This leads to a distinction between linear functions, homogeneous functions, homogenous linear functions, bilinear functions, and multilinear functions. Each of them is defined in accessible terms and illustrated with examples. We analyze traditional, matrix-based, LCA, and conclude that LCA is not linear in any of the senses defined. Despite the negative answer to the research question, there are many respects in which LCA can be regarded to be, at least to some extent, linear. We discuss a few of such cases. We also discuss a few practical implications for practitioners of LCA and for developers of new methods for LCI and LCIA.

中文翻译：

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主流 LCA 是线性的吗？

文献中经常提到 LCA 是线性的，没有证明，甚至没有明确定义线性标准。在这里，我们研究术语线性的含义，以及与此相关的 LCA 是否确实是线性的问题。我们在数学模型的上下文中探索术语线性的不同含义。这导致了线性函数、齐次函数、齐次线性函数、双线性函数和多重线性函数之间的区别。它们中的每一个都以易于理解的术语进行定义，并通过示例进行了说明。我们分析了传统的、基于矩阵的 LCA，并得出结论，LCA 在任何定义的意义上都不是线性的。尽管对研究问题的回答是否定的，但在许多方面 LCA 可以被视为，至少在某种程度上，是线性的。我们讨论了一些这样的案例。我们还讨论了一些对 LCA 实践者和 LCI 和 LCIA 新方法开发者的实际意义。