I.The Basic Information of the Course
Course Number:202012420012
The English Name of the Course：Combinatorial Matrix Theory
The Chinese Name of the Course：组合矩阵论
In-classHours andAllocation:total class hours:32, classroom teaching: 28 classhours, classroom discussion: 4 classhours
Credit(s):2
Semester:1
AppliedDiscipline(Professional Degree Category)：Mathematics
CourseObjectOriented：Academic Doctor，Academic Master
EvaluationMode:Process Evaluation.
TeachingMethod:Blended Teaching
CourseOpening Department: School of Mathematical Sciences
II.Prerequisite Course
Graph Theory, Advanced Algebra
III. The Objectives and Requirements of the Course
The course is aCombinatorial Matrix Theory.The aim is to cultivatethe ability of students to use matrix theory and linear algebra method to prove the combinatorial theorem and describe and classify the combinatorial structure, at the same time, the ideas and demonstration methods of combinatorial theory are also used for the precise analysis of matrix and to reveal the inherent combinatorial properties of matrices.
By learning this course, the students should reach the following basic requirements:matrix decomposition, matrix generalized inverse, nonnegative matrix, graph and matrix, generalized inverse sign pattern and so on.
IV. The Content of the Course
Chapter 1 Introduction (6 class hours)
1.1 Special matrices
1.2 Polynomials of matrices
1.3 Decomposition of matrices
Chapter 2 Generalized inverse of matrices（6 class hours）
2.1 {1}-inverse of matrices and its application in linear systems
2.2 Moore-Penrose inverse of matrices and its application in linear system
2.3 Generalized spectral inverse of matrices
Chapter 3 Nonnegative matrices (6 class hours)
3.1 Perron Frobenius theorem
3.2 Primitive matrices and nonprimitive matrices
3.3 Specialnonnegativematrices
Chapter 4 Matrix and graph(8 class hours)
4.1 Graph theory
4.2 Matrix of graph
4.3 Resistance distance
4.4Random walk
Chapter 5 Sign pattern of generalized inverse (6 class hours)
5.1 Sign nonsingular mode
5.2 Matrix inverse sign pattern
5.3 Generalized inverse sign pattern
Ideological and political cases: the course of combinatorial matrix theory is rich in content and profound in research methods. It is a very practical mathematics course and plays a significant role in promoting the development of students' ability to apply mathematics and innovative thinking. In the course of teaching, the following two cases are introduced:
Case 1: Development of matrix theory
According to the development of mathematics in the world, the concept of matrix appears in the 1850s for the need of solving linear equations. In 1850, a British mathematician Sylvester introduced the concept of matrix because he couldn't use determinant when he studied linear systems with different number of equations and unknown quantities. In 1855, the British mathematician caylag introduced the concept of matrix for the sake of simplicity and convenience when he studied the invariants under linear transformation. However, the matrix had already sprouted in China before BC. Matrix has been described in "Nine Chapters of Arithmetic" in China, but it has not been studied as an independent concept, but only used to solve practical problems, so it has not been able to form an independent matrix theory. The theory of matrix develops very fast. By the end of the 19th century, matrix theory has been basically formed. By the 20th century, matrix theory has been further developed. At present, it has developed into a branch of mathematics which has been widely used in physics, cybernetics, robotics, biology, economics and other disciplines. This also reflects the highly developed mathematics in ancient China.
Case 2: The spirit of mathematical science
The five thousand years’ development of mathematics is a history of continuous development, perseverance, exploration and innovation of mathematics workers on the basis of inheritance, and a history of continuous pursuit of rational truth, structural perfection and practical application. In this process, the distinct spirit of mathematical science is bred. The so-called mathematical spirit refers to the essential characteristics of the humanistic social value of the mathematical science itself, as well as the scientific attitude and scientific spirit of the humanistic social value embodied by generations of mathematicians in the process of exploring the mysteries of the mathematical science and promoting the development of the mathematical science. On the road of scientific research and innovation, the cultivation of graduate students has the spirit of seeking truth from facts, persistently pursuing truth, and striving for perfection; the spirit of being particularly rigorous, meticulous and self-improvement; the spirit of constant innovation and scientific inclusiveness.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1. Zhan Xingzhi, Matrix Theory, Higher Education Press, Beijing, 2008.
2. Liu Bailian, Combinatorial Matrix Theory, Science Press, Beijing, 1998.
3. R. A. Brualdi, H J. Ryser，Combinatorial Matrix Theory, Cambridge University Press, 1991.
4. Ramon, Garcia, Roger A. horn, translated by Zhang Mingyao, Matrix Theory and Application, China Machine Press, 2020
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