1、Basic course information
Course number:202032020004
Course Chinese Name:矩阵论
Course English Name:Matrix Theory
Class hours and distribution:Total class hours: 32, 30 in class, 2 in class discussion
CATS points:2
Semester:1st
Applicable disciplines (professional degree category):Science and Engineering
Object oriented:academic master, professional master
Assessment method:closed-book exam
Teaching method:classroom teaching
Course starter:College of Mathematical Sciences
2、Prerequisite courses
Linear algebra, advanced mathematics
3、Course objectives and basic requirements
Matrix theory is a follow-up course of linear algebra. On the basis of linear algebra, it further introduces linear space and linear transformation, Euclidean space and unitary space, as well as the linear transformation on these space, and profoundly reveals the essence and thought of linear transformation on finite dimensional space. In order to expand the field of advanced mathematics analysis, the theory of matrix analysis is constructed by introducing vector norm and matrix norm. At the same time, this course also introduces ideological and political cases into the classroom, so that students can understand the application of this course in the construction of military national defense while mastering the relevant knowledge of mathematical theory.
From an application perspective, matrix algebra are an important foundation for numerical analysis, and an important tool for studying linear dynamic systems. For the practicality of matrix theory, for the calculation problem of matrix algebra and analysis, Matlabis used to realize quick calculation and analysis.
4、Teaching content
This course simplifies the contents of the original course that are relatively backward and intersected with undergraduate and graduate students, and increases the basic theory of mathematics necessary for modern science. It changes the teaching thought that emphasizes theory over practice. By "improving the starting point, strengthening the foundation, dividing the levels, and rationalizing the relationship", both the order of knowledge structure and the interrelationship of knowledge modules are considered, highlighting the teaching content of the matrix theory course.Combining Ideological and Political Cases with Matrix Theory Course Teaching, on the one hand, it can enhance students' understanding of the knowledge learned, and on the other hand, it can also guide students to apply the knowledge learned, especially some specific technical issues in military operations. Itcannot only exemplify the application of matrix theory, but also provide patriotism education on national defense awareness for students, and use typical matrix theory knowledge to solve military problems.Through the teaching of example questions, students' strong patriotism is stimulated. At present, the United States and its allies frequently carry out targeted military exercises in South China Sea and the Pacific Ocean, which seriously threatens our sovereignty over the territorial waters.Only by scientifically solving many similar problems can they understand the profound meaning of loving their motherland.
1. Linear space and linear transformation
Understand the concept of linear space, master the formulas of base transformation and coordinate transformation; grasp the subspace and dimension theorem, understand the meaning of isomorphism in linear space; understand the concept of linear transformation, master the matrix representation of linear transformation, and apply the knowledge learned to solveproblems in real life.
2. Inner product space
Understand the concept of inner product space, grasp the orthogonal relationship between the orthogonal basis and subspace; understand the meaning of the isomorphism of inner product space, and master the method of judging the orthogonal transformation; understand the concept of chieftain space,candetermine whether a space isthe method of chieftain space, master the similarities and differences of chieftain space and real inner product space; master the concept and decision theorem and nature of normal matrix, and give the application of ideological and political cases.
3. Diagonalization of matrix andJordan canonical form
Master the method of judging the similar diagonalization of matrix; understand the meaning of Hermitian quadratic form.Can findthe Jordan standard form of the matrix;can findthe Smith standard form;can findthe Jordan standard form, andcanuse the Jordan standard form to solve applications in real life.
4.Matrix factorization
Canfind the triangular decomposition and UR decomposition of the matrix; full rank decomposition and spectral decomposition of the simple matrix; understand the singular value and extreme decomposition of the matrix.
5. Important digital features of vector and matrix
Understand the vector norm and matrix norm; the equivalence of vector norm in finite dimensional linear space; the compatibility of vector norm and matrix norm.Canuse mathematical thoughts and thinking to solve the problem of proof of norm.
6.Matrix analysis
Understand the concept of the limits of vector and matrix; master the method of judging the convergence of the power series of matrix; understand the Kronecker product of matrix; know the differentiation and integration of matrix.
5、Main reference materials, references and learning resources of the course
(1) Textbooks, monographs and references
1. Lin Mn, Wu Hongmei. Linear space and matrix theory. Harbin University of Engineering Press, 2016.
2. Lin Mn. Matrix theory. National Defense Industry Press, 2013.
Ideological and political cases
Teaching content |
Basic elements or cases |
Equipment damage simulation |
In the battlefield, especially in the modern war, equipment is faced with various kinds of damages, such as a large number of broken pieces formed after missile explosion. How to use the Euclidean distance and geometric ray method in inner space to judge equipment damage in the battle simulation and battle test?Using matrix and geometry model and its hit judgment basis, and then using Monte Carlo method to simulate the core model of fragment operation can provide important basis for damage assessment.This problem gives a simulation problem by proposing a mathematical problem, the establishment of a coordinate system and the geometric description of equipment, the basis of equipment judgment in fragmentation, and the generation of killing rays, etc.In this way, students can be given a patriotism education in national defense awareness. It will also stimulate students' strong patriotism and make them understand the profound meaning of loving the motherland. |
Using matrix to make hill password |
Cryptography plays an important role in economy and military.In cryptography and cryptanalysis, we can choose different invertible matrices, different mapping relations, and arrange the numbers corresponding to letters to get different matrices. In this way, there are many ways of encryption and decryption, so as to ensure the confidentiality of information transmission.Through the application of matrix multiplication and inverse matrix, the calculation of matrix decomposition and solving equations and other mathematical knowledge is incorporated into the calculation and decryption of ciphers, and then applied to the military and other aspects, reflecting the powerful application of matrix theory. |
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Signature of department leader in charge of teaching:
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