This set of notes re-proves known results on weighted automata (over a field, also known as multiplicity automata). The text offers a unified view on theorems and proofs that have appeared in the literature over decades and were written in different styles and contexts. None of the results reported here are claimed to be new. The content centres around fundamentals of equivalence and minimization, with an emphasis on algorithmic aspects. The presentation is minimalistic. No attempt has been made to motivate the material. Weighted automata are viewed from a linear-algebra angle. As a consequence, the proofs, which are meant to be succinct, but complete and almost self-contained, rely mainly on elementary linear algebra.

中文翻译：

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关于加权自动机的等价和最小化的注记

这组笔记重新证明了加权自动机（在一个领域，也称为多重自动机）上的已知结果。本书对数十年来出现在文献中并以不同风格和背景编写的定理和证明提供了统一的观点。这里报告的结果都没有声称是新的。内容围绕等价和最小化的基本原理，重点放在算法方面。演示文稿是简约的。没有试图激发材料。从线性代数的角度来看加权自动机。因此，本应简洁但完整且几乎自成一体的证明主要依赖于基本线性代数。