The Matrix Cookbook.pdf

矩阵知识的工具书，建议查询，不建议系统学习。 What is this? These pages are a collection of facts (identities, approximations, inequalities, relations, ...) about matrices and matters relating to them. It is collected in this form for the convenience of anyone who wants a quick desktop reference . Disclaimer: The identities, approximations and relations presented here were obviously not invented but collected, borrowed and copied from a large amount of sources. These sources include similar but shorter notes found on the internet and appendices in books - see the references for a full list. Errors: Very likely there are errors, typos, and mistakes for which we apologize and would be grateful to receive corrections at cookbook@2302.dk. Its ongoing: The project of keeping a large repository of relations involving matrices is naturally ongoing and the version will be apparent from the date in the header. Suggestions: Your suggestion for additional content or elaboration of some topics is most welcome acookbook@2302.dk. Keywords: Matrix algebra, matrix relations, matrix identities, derivative of determinant, derivative of inverse matrix, differentiate a matrix. Acknowledgements: We would like to thank the following for contributions and suggestions: Bill Baxter, Brian Templeton, Christian Rishøj, Christian Schr¨oppel, Dan Boley, Douglas L. Theobald, Esben Hoegh-Rasmussen, Evripidis Karseras, Georg Martius, Glynne Casteel, Jan Larsen, Jun Bin Gao, Ju¨rgen Struckmeier, Kamil Dedecius, Karim T. Abou-Moustafa, Korbinian Strimmer, Lars Christiansen, Lars Kai Hansen, Leland Wilkinson, Liguo He, Loic Thibaut, Markus Froeb, Michael Hubatka, Miguel Bar˜ao, Ole Winther, Pavel Sakov, Stephan Hattinger, Troels Pedersen, Vasile Sima, Vincent Rabaud, Zhaoshui He. We would also like thank The Oticon Foundation for funding our PhD studies. 《矩阵食谱》是一本关于矩阵知识的工具书，它汇集了各种矩阵及其相关主题的事实，包括恒等式、近似值、不等式和关系等。这本书的主要目的是为需要快速查阅矩阵知识的人提供方便。书中所包含的内容并非原创，而是从大量来源收集、借用和复制的，这些来源包括互联网上的简短笔记和书籍附录。尽管如此，书中难免存在错误、打字错误或错误，作者对此表示歉意，并欢迎读者通过指定邮箱(cookbook@2302.dk)提供更正。 该书的更新是持续进行的，版本号会显示在页眉，以便读者了解最新状态。作者鼓励读者提出增加内容或深入探讨某些话题的建议，可以通过电子邮件(acookbook@2302.dk)发送。书中的关键词包括矩阵代数、矩阵关系、矩阵恒等式、行列式导数、逆矩阵导数以及矩阵微分。 感谢以下人员对书中的贡献和建议：Bill Baxter、Brian Templeton、Christian Rishøj等多位人士。同时，也要感谢奥梯信基金会(The Oticon Foundation)对作者博士研究的资助。 书中的内容涵盖了广泛的矩阵理论和应用，例如： 1. **矩阵代数**：涉及矩阵的基本运算，如加法、减法、乘法和标量乘法。还包括矩阵的转置、逆矩阵、迹（对角元素之和）和行列式。 2. **矩阵关系**：讨论了矩阵之间的关系，如秩、奇异值分解、特征值和特征向量等。 3. **矩阵恒等式**：介绍了关于矩阵运算的恒等式，如克拉默法则、高斯消元法、矩阵乘积的性质等。 4. **矩阵的导数**：讲解了如何对矩阵的行列式和逆矩阵进行微分，这对于线性代数和偏微分方程的求解至关重要。 5. **矩阵微分**：涉及矩阵函数的微分，这在优化问题和控制系统理论中非常重要。 6. **不等式**：书中可能包含了矩阵范数不等式和其他关于矩阵大小的比较。 这本书是矩阵计算和应用领域的宝贵资源，对于科研人员、工程师和学生来说，是一个方便快捷的参考工具。尽管不建议作为系统学习矩阵论的教材，但对于解决特定问题或查证公式，它无疑是一个非常实用的资料库。