Readership: Graduate students, practitioners and researchers interested in calculus, matrices, optimisation problems, statistical models and/or their applications.

The book has become the standard reference on matrix differential calculus since first published in 1988. This is the third edition of the book, with seven parts devoted to the theory and application of matrix differential calculus. Matrix differential calculus was pioneered and promoted by the authors and developed on with many other contributors in the last several decades, especially the late Professor Heinz Neudecker. As stated in the preface to this new edition, ‘Heinz Neudecker must be regarded as its founding father’.

The first part is an introductory review on matrices. Chapter 1 includes basic definitions and results such as matrix addition and multiplication, rank, inverse, determinant, trace, eigenvalue and eigenvector, Schur's decomposition theorem and singular‐value decomposition. Chapters 2 and 3 cover selected, useful results for later chapters, involving Kronecker product, Hadamard product, vec operator, Moore‐Penrose inverse, commutation matrix and duplication matrix.

The second part is on differentials in theory. Chapter 4 introduces the mathematical preliminaries. Chapters 5–7 cover the uniqueness and existence of the differential, the first and second identification theorems, the mean‐value theorem, remarks on notation, the chain rule, partial derivatives, the Hessian matrix, unconstrained and constrained optimisation, necessary and sufficient conditions for a local minimum without or under constraints, three versions of the implicit function theorem, among others which are fairly fundamental.

The third part explores differentials in practice. It principally shows how to implement the theory and use important differentials with definitions, rules, propositions, theorems with proofs and provided examples. The first and second identification theorems incorporate Jacobians and Hessian matrices for scalar, vector and matrix functions of scalar, vector and matrix variables. The differentials with Jacobian and Hessian matrices are presented extensively, including matrix trace, determinant and eigenvalue functions, the eigenvector function, as well as Kronecker and Hadamard products.

The fourth part is a collection of some key inequalities, including the Cauchy–Schwarz inequality, its matrix versions (matrix determinant and trace versions), Fischer's min–max theorem, Poincarè separation theorem, Hadamard's inequality, Karamata's inequality, Hölder's inequality, Minkowski's inequality, Schlömilch's inequality, the least squares and generalised least squares with their restricted counterparts.

Part five discusses several topics in linear model. Chapter 12 overviews the statistical preliminaries, cumulative distribution function, joint density function, expectation, variance and covariance, independence of random variables and the normal distribution. Chapters 13 and 14 present the most important topics in the theory of linear models, namely, the Gauss–Markov theorem, the methods of ordinary, generalised and restricted least squares, Aitken's theorem, estimable functions, linear constraints, singular variance matrix, best quadratic unbiased and invariant estimators of the variance, best linear unbiased predictors, and local sensitivity of the posterior mean and precision.

Part six applies the maximum likelihood methods to various models and issues and especially the full and limited information maximum likelihood methods used in simultaneous equations. The topics in one‐, two‐ and multi‐mode component analysis, principal components, factor analysis, canonical correlation, correspondence analysis and linear discriminant analysis are widely used in psychometrics and multivariate statistics. Chapter 17 provides additional examples applying the various results obtained, including those on Kronecker and Hadamard products, eigenvalues and eigenvectors, matrix determinant and trice optimisation problems and matrix calculus.

Part seven offers a summary of the essentials of matrix calculus. It has a basic introduction followed by a coverage on the useful definitions and results for differentials, chain rule for differentials, vector calculus, matrix calculus, Kronecker product and vec operator, commutation and duplication matrices, least squares and maximum likelihood, and so forth, with examples.

This edition continues to be successful in maintaining those distinguishing features in the previous editions. The book is well written and lucid. It is self‐contained. It builds on good notation, a unique approach of using differentials (rather than derivatives) and the vectorial operation amidst the chain rule procedure. It covers the theory and a wide range of topics with applications in not only statistics, econometrics and psychometrics but also related areas in biosciences, social and behavioural sciences and many others. The proofs of many key theorems are both rigourous and easy to follow. The examples and exercises are extremely helpful, with newly added exercises in this edition linking to research problems and requests from readers. The bibliographical notes are informative for understanding the relevant literature and especially beneficial for those readers interested in further study or research.

This edition has updated materials and references throughout the book, especially in sections involving matrix functions, complex differentiation, Jacobians of transformations, differentiation of eigenvalues and eigenvectors and Hessian matrices for scalar functions. It has made two new sections on correspondence analysis and linear discriminant analysis, respectively. Obviously, a new chapter at the end of the book presents a collection of the essential definitions and results for matrix differential calculus. It is devoted to a practical yet self‐contained and handy summary of the whole book and can be studied by readers independently from the rest of the text, that is, without going into theoretical details. Some pertinent extensions are absent, such as the Khatri–Rao product (related to Sections 2.2 and 3.6), the matrix versions of the Cauchy–Schwarz and Kantorovich inequalities (in the Löwner partial ordering) and their applications in efficiency comparisons (related to Sections 11.2, 11.3, 11.29–11.32, 13.18 and 13.19), the local sensitivity of generalised least squares (related to Sections 14.15 and 14.16) and the pseudo likelihood estimation (related to Sections 15.2 and 15.3).

As proven by various applications in different areas and as noted in the preface to the third edition, the technique of matrix calculus through differentials ‘is still a remarkably powerful tool’. Recently, matrix differential calculus has been found to be useful (as we should expect of such a powerful tool) in data analytic areas like deep learning and even ecological areas like matrix population models. More concrete uses in newer areas could be explored in the next edition.

In summary, this book is highly recommended as an advanced text or a reference handbook to anyone who uses or is interested in matrices, calculus, optimisation problems, statistical models and/or their applications in individual studies.

读者群：对微积分，矩阵，优化问题，统计模型和/或其应用感兴趣的研究生，从业人员和研究人员。

自1988年首次出版以来，这本书已成为矩阵微积分的标准参考。这是本书的第三版，其中七个部分专门介绍矩阵微积分的理论和应用。矩阵微分学由作者开创和推广，并在过去的几十年中与许多其他贡献者一起发展，尤其是已故的海因茨·诺德克教授。如新版序言所述，“必须将亨氏·诺德克（Heinz Neudecker）视为其创始之父”。

第一部分是对矩阵的介绍性回顾。第1章包括基本定义和结果，例如矩阵加法和乘法，秩，逆，行列式，迹线，特征值和特征向量，舒尔分解定理和奇异值分解。第2章和第3章介绍了后续章节中选定的有用结果，涉及Kronecker乘积，Hadamard乘积，vec运算符，Moore-Penrose逆，交换矩阵和重复矩阵。

第二部分是理论上的差异。第4章介绍数学预备知识。第5–7章介绍了微分的唯一性和存在性，第一和第二个辨识定理，均值定理，关于符号的注释，链式规则，偏导数，Hessian矩阵，无约束和约束优化，必要条件和充分条件对于无约束或有约束的局部最小值，隐式函数定理的三个版本以及其他相当基本的版本。

第三部分探讨了实践中的差异。它主要显示了如何实施该理论以及如何使用具有定义，规则，命题，定理的重要微分以及证明和提供的示例。第一和第二个识别定理将标量，向量和矩阵变量的标量，向量和矩阵函数合并到Jacobian和Hessian矩阵中。广泛介绍了Jacobian和Hessian矩阵的微分，包括矩阵轨迹，行列式和特征值函数，特征向量函数以及Kronecker和Hadamard乘积。

第四部分是一些关键不等式的集合，包括Cauchy-Schwarz不等式，其矩阵形式（矩阵行列式和迹线形式），Fischer的最小-最大定理，Poincarè分离定理，Hadamard的不等式，Karamata的不等式，Hölder的不等式，Minkowski的不等式。 ，Schlömilch不等式，最小二乘法和广义最小二乘法以及受限制的对应项。

第五部分讨论线性模型中的几个主题。第12章概述了统计初步数据，累积分布函数，联合密度函数，期望，方差和协方差，随机变量的独立性和正态分布。第13章和第14章介绍了线性模型理论中最重要的主题，即高斯-马尔可夫定理，普通，广义和受限最小二乘法，Aitken定理，可估计函数，线性约束，奇异方差矩阵，最佳二次方方差的无偏和不变估计量，最佳线性无偏预测值以及后均值和精度的局部敏感性。

第六部分将最大似然法应用于各种模型和问题，尤其是联立方程中使用的全部和有限信息最大似然法。一模，二模和多模成分分析，主成分，因子分析，典范相关性，对应性分析和线性判别分析中的主题广泛用于心理计量学和多元统计中。第17章提供了应用获得的各种结果的其他示例，包括有关Kronecker和Hadamard乘积，特征值和特征向量，矩阵行列式和三次优化问题以及矩阵演算的结果。

第七部分总结了矩阵演算的要点。它有一个基本的介绍，然后介绍了微分的有用定义和结果，微分的链规则，矢量微积分，矩阵微积分，Kronecker乘积和vec运算符，交换和复制矩阵，最小二乘和最大似然等，有例子。

该版本继续成功维护了先前版本中的那些独特功能。这本书写得很清楚。它是独立的。它建立在良好的表示法，使用差异（而不是导数）的独特方法以及链式规则过程中的矢量运算的基础上。它涵盖了理论和广泛的主题，不仅适用于统计学，计量经济学和心理计量学，还涉及生物科学，社会和行为科学以及许多其他领域的相关领域。许多关键定理的证明既严格又易于遵循。这些示例和练习非常有帮助，此版本中新增的练习可以链接到研究问题和读者的要求。

此版本在整本书中都有更新的材料和参考，特别是涉及矩阵函数，复数微分，变换的雅可比行列，特征值和特征向量的微分以及标量函数的Hessian矩阵的章节。它分别在对应分析和线性判别分析上增加了两个新的部分。显然，这本书结尾的新章节提出了矩阵微积分的基本定义和结果的集合。它致力于整本书的实用且自成一体且方便的摘要，读者可以独立于本书的其余部分进行研究，也就是说，无需深入理论。缺少一些相关的扩展，例如Khatri–Rao产品（与2.2和3.6节相关），

正如在不同领域的各种应用所证明的那样，以及在第三版的序言中指出，通过微分进行矩阵演算的技术“仍然是非常强大的工具”。最近，发现矩阵微分演算在数据分析领域（如深度学习）甚至生态领域（如矩阵人口模型）中很有用（我们应该期望这种强大的工具）。在下一版中将探讨在较新领域中更具体的用途。

总之，强烈建议将本书作为高级教科书或参考手册，以供使用或对矩阵，微积分，优化问题，统计模型和/或它们在个别研究中的应用感兴趣的任何人使用。