青岛大学研究生课程教学大纲

课程编号：  4472012 课程名称：  矩阵理论  

( Course No.  4472012 : Course Name  Matrixtheory  )

一、课程简介（Description）             

1. 1.   学分/学时（Credit/Hours）：3/48

2. 2.   开课学期（Semester）：1

3. 3.   内容简介（Introduction）：

《矩阵理论》课程是面向理工科专业研究生重要的基础理论课程。课程旨在以学生已掌握的线性代数和高等数学知识为基础，使其进一步熟练掌握关于矩阵基本的、核心的理论知识，同时深刻领会其中的思想与方法，为在今后的学习和研究中熟练运用矩阵理论和方法奠定坚实的基础。

课程内容主要包括线性空间与线性变换、矩阵范数、矩阵分析及其应用、矩阵分解、特征值的估计、广义逆矩阵、若干特殊矩阵及其应用等。重点介绍工程技术、经济管理中应用较多的矩阵分析基础理论和基本方法，有很强的针对性和目的性。

(“Matrix Theory” is an important basic theory coursefor postgraduates majoring in science and engineering. Based on the knowledgeof linear algebra and higher mathematics the students have mastered, the coursemakes them master the basic and core theoretical knowledge and algorithms onthe matrix, deeply comprehend the ideas and methods of the matrix, and lay asolid foundation for the skillful use of matrix methods in future study andresearch.

The contents of this courseinclude linear space and linear transformation, matrix norm, matrix analysis,matrix function and its application, matrix decomposition, estimation ofeigenvalues, generalized inverse matrices, some special matrices and their applications.It mainly introduces the basic theories and methods of matrix analysis extensivelyapplied in engineering technology and economic management, which are targetedand purposeful.)

1. 4. 教材名称（Textbooks）：矩阵论（第4版），程云鹏等著，西北工业大学出版社，2013年(Matrix theory (The 4^th Edition), YunpengCheng, Northwestern Polytechnical University press, 2013)

2. 5. 先修课程（Prerequisites）：高等数学，线性代数(higher mathematics, linearalgebra)

3. 6. 授课对象（Teaching objects）：硕士一年级研究生(Graduate students in the firstgrade)

4. 7. 适合专业（Suited Professions）：控制科学与工程(Control Science and Engineering)

5. 8. 教学语言（Language of Instruction）：中文(Chinese)

6. 9. 参考文献（References）：

(1) 矩阵分析与应用，张贤达著，清华大学出版社，2004年(Matrix analysis and applications, Xianda Zhang, TsinghuaUniversity press, 2004)

(2) 系统与控制理论中的线性代数，黄琳，科学出版社，1984年(Linear algebra in systems and control theories, LinHuang, Science Press, 1984)

(3) 矩阵分析，杨奇译，天津大学出版社，1998年(Matrix Analysis. Roger A. Horn, Charles R. Johnson.Cambridge Press, 1994)

1. 10.任课教师及团队（Coordinators & Team）：纪志坚刘华波刘振韩乔妮王小成 (Zhijian Ji, Huabo Liu, Zhen Liu, Qiaoni Han, XiaochengWang)

2. 二、教学目标（Educational Objectives）

        《矩阵理论》课程是面向理工科专业研究生重要的基础理论课程。通过本课程的学习，使学生掌握矩阵分析相关的基本概念、基础理论和基本运算，全面理解矩阵范数、矩阵分析、矩阵分解、特征值估计、广义逆矩阵等基础理论及其应用，为今后在控制科学与工程专业的进一步学习和研究打下扎实的基础。

通过本课程中基本概念和基本定理的阐述与论证，培养理工科硕士研究生的抽象思维与逻辑推理能力，提高学生的数学理论素养。在重视数学论证的同时，强调数学概念的物理背景，培养学生应用数学知识解决实际工程技术问题的能力。

(Matrix theory is an important basic theory course forgraduates majoring in science and engineering. Through the study of thiscourse, students can grasp the basic concepts, basic theories and basicoperations of matrix theory, and fully understand the basic theories andapplications of matrix norms, matrix analysis, matrix decomposition, eigenvalueestimation, generalized inverse matrix and so on, and lay a solid foundation for further study and research in control science andengineering.

The abstract thinking and logical deduction ability ofthe graduate students major in science and engineering are trained and thestudents' mathematical theoretical attainments are improved through theexposition and demonstration of the basic concepts and basic theorems in thiscourse. While emphasizing mathematical reasoning, it emphasizes the physicalbackground of mathematical concepts, and trains students' ability to applymathematical knowledge to solve practical engineering problems.)

1. 1. 目标1（Objective 1）：理解线性空间的相关概念，掌握线性变换及其矩阵表示，掌握欧式空间的相关概念。(Understand the related concepts of linear space, graspthe linear transformation and its matrix expression, and grasp the relatedconcepts of Euclidean space.)

2. 2. 目标2（Objective 2）：掌握向量范数、矩阵范数、诱导范数的概念及其计算。(Master the concepts of vector norm, matrix norm, inducednorm and their computations.)

3. 3. 目标3（Objective 3）：掌握矩阵序列、矩阵级数的概念，能熟练求取矩阵序列的极限及对矩阵幂级数敛散性进行判定；掌握矩阵函数的表示及其相应的计算方法。(Master the concepts of matrix sequence and matrix series,obtain the limit of matrix sequence and determine the convergence anddivergence of matrix power series; master the representation of matrix functionand its corresponding calculation methods.)

4. 4. 目标4（Objective 4）：掌握矩阵的三角分解、QR分解、满秩分解、奇异值分解等。(Master the triangular decomposition, QR decomposition,full rank decomposition, singular value decomposition and so on.)

5. 5. 目标5（Objective 5）：理解特征值界的估计，掌握Gerschgorin圆盘定理及其推广，理解Hermite矩阵特征值的表示，理解广义特征值问题。(Understand the estimation of eigenvalue bounds, master theGerschgorin disk theorem and its generalization, understand the representationof eigenvalues of Hermite matrix, and understand generalized eigenvalueproblems.)

6. 6. 目标6（Objective 6）：掌握广义逆和伪逆矩阵的概念和计算方法，能利用此概念对线性方程组进行求解和理论分析。(Master the concept and calculation method of generalizedinverse and pseudo inverse matrix, use the concepts to solve linear equations andmake theoretical analysis.)

7. 7. 目标7（Objective 7）：理解若干特殊矩阵的形式和特点，能利用其特殊性进行相应的分析与计算。(Understand the forms and characteristics of some specialmatrices, take advantage of their particularities to make analysis and calculations.)

8. 三、教学内容/学时分配/授课方式（Topics Covered/Credit Hours/Lecture Model）

四、考评方法（Evaluation Method）

可采取（但不局限于）以下方式进行课程学习效果考核(We can adopt (but not limited to) the following toassessthe effectiveness of learning)：

1. 1. 作业（Homework）：认真及时完成作业。(Finish the homework carefullyand in time.)

2. 2. 研究项目（Major Project）：无。(Null.)

3. 3. 书面报告（Written Reports）：无。(Null.)

4. 4. 口头报告（Oral Reports）：严谨的逻辑主线，清晰流利的表达。(Rigorous logic line, clear and fluent expression.)

5. 5. 课内实验（Experiments）：积极踊跃参与实验，认真负责完成实验。(Actively participate in the experiments, complete theexperiment conscientiously and responsibly.)

6. 6. 期末考试（Exams）：独立认真完成期末考试。(Finish the final examinations independently andconscientiously.)

7. 五、成绩比例（Grading Scale）

针对课程所采用的各类考评办法，给出以百分计算的考评成绩及其在总成绩中所占的比例，最终给出该门课程的学习成绩（以百分计）。(In view of the various evaluation methods used in thecourse, the results of the evaluation and the proportion in the total resultsare given, and the results of the course are finally provided.)

六、预期效果（Assessment of Learning Outcomes）

针对第二项所设置各个教学目标，通过上述内容的学习和所采取的考评方法，预估学生学完本门课程后所能掌握和了解的知识、方法和技能，以及所具备的具体能力等。(According to the teaching objectives set up in thesecond items, the knowledge, methods and skills that the students can masterand understand after they have finished the course are estimated through thestudy of the above content and the method of evaluation.)

1. 1.      目标1（Objective 1）：通过课程学习，掌握线性空间和欧式空间的相关概念，掌握线性变换及其矩阵表示的方法。(Through the course learning, grasp the concepts of linear space andEuclidean space, and grasp the methods of linear transformation and matrixrepresentation.)

2. 2.      目标2（Objective 2）：学生掌握了向量范数、矩阵范数、诱导范数的概念且能够进行计算求解。(Students have mastered the concepts of vector norm,matrix norm and induction norm, and can solve them.)

3. 3.      目标3（Objective 3）：根据掌握的基本概念，能熟练求取矩阵序列的极限以及判定矩阵幂级数的敛散性等；掌握矩阵函数的表示方法及计算方法。(According to the mastered basic concepts, the limit ofmatrix sequence can be obtained and the convergence and divergence of the powerseries of matrix can be determined, and the representation method andcalculation method of matrix function can be grasped.)

4. 4.      目标4（Objective 4）：通过课程学习，学生能够根据需要对矩阵进行三角分解、QR分解、满秩分解、奇异值分解等。(Through the course learning, students can performtriangular decomposition, QR decomposition, full rank decomposition, singularvalue decomposition and so on.)

5. 5.      目标5（Objective 5）：通过课程学习，学生能够掌握特征值界的估计、掌握Gerschgorin圆盘定理及其推广、理解Hermite矩阵特征值的表示。(Through the course learning, students can grasp theestimation of eigenvalue bounds, grasp the Gerschgorin disk theorem and its expansion,and understand the representation of eigenvalues of Hermite matrix.)

6. 6.      目标6（Objective 6）：学生掌握广义逆和伪逆矩阵的概念和计算方法，能利用此概念对线性方程组进行求解和理论分析。(Students can grasp the concept of generalized inverseand pseudo inverse matrix, and use these concepts to solve the linear equationsand make a theoretical analysis.)

7. 7.      目标7（Objective 7）：通过课程学习，能够利用特殊矩阵的形式特点进行相应的分析与计算。(Through the course learning, students can take advantageof the characteristics of some special matrices to make analysis and calculations.)

1、矩阵论（第4版），程云鹏等著，西北工业大学出版社

2、矩阵论简明教程（第2版）徐仲等

3、矩阵分析与应用，张贤达著，清华大学出版社，2004年(Matrix analysis and applications, Xianda Zhang, TsinghuaUniversity press, 2004)

4、系统与控制理论中的线性代数，黄琳，科学出版社，1984年(Linear algebra in systems and control theories, LinHuang, Science Press, 1984)