Matrix Analysis and its Applications
Spring 2018

"I did not look for matrix theory. It somehow looked for me."

--Olga Taussky Todd in American Mathematical Monthly

Instructor: Professor Lixin Yan (mcsylx@mail.sysu.edu.cn)

Office: Room 702 (School of Mathematics, Sun Yat-Sen University)

Contact: 020－84110123

Teaching Assistant: Yikun Zhang (yikunzhang@foxmail.com)

TA Office Hour: By appointment

Lecture Time and Location:

Thursday Section 5-8 ( 14:20--18:00 )

Lectures will be held in Room 415, the new building of the School of Mathematics.

Course Description:

This is the second honor course for elite sophomores of the Applied Mathematics group at Yat-sen Honor College. As what we did in the first course (Fourier Analysis and its Applications), the enrolled student will continue lecturing to the class in a pre-assigned order.

The lecture will follow the main stream of the textbook. (See below for the intended textbook.) A year-long exposure to Linear Algebra will undoubtedly facilitate your lecture preparation and finish homework assigments. However, considering the fact that this honorable textbook covers essentially all the topics in Matrix Analysis, students are encouraged to resort to the Internet or other relevant books in order to smooth their lectures. Feel free to ask the instructor, teaching assistant, and other fellow students for help!

Textbook:

Matrix Analysis, Second Edition (Author: Roger A. Horn & Charles R. Johnson)
A reprint edition is published by Posts & Telecom Press and available on Amazon, China. The original price is 99 RMB per book, while some discounts would be made by Amazon.

Grading:

80% (Lecture performance and evaluation from fellow classmates) + 20% (Final Reading Report)

Syllabus:

Due to the intense schedule, we are unable to cover all the materials of the textbook. Hopefully, we can begin Chapter 4 at the end of this semester.

Week Lecturer Textbook Notes
4 Qiwen Zhou & Yue Hu Sec 1.0, 1.1, 1.2 Lecture 1 Notes (by Yikun Zhang)
5 Tomb-sweeping Day (No Class)
6 Sisi Gai & Dan He Sec 1.3, 1.4 Lecture 2 Notes & Matlab Code for the Power Iterative Algorithm (by Yikun Zhang)
Further readings: Paper 1: Short proofs of theorems of Mirsky and Horn on diagonals and eigenvalues of matrices ;
Paper 2: Matrices with prescribed off-diagonal elements
7 Jian Yao & Minghan Dai Sec 1.4, 2.1 Lecture 3 Notes (by Yikun Zhang)
8 Yanran Li, Ruicheng Li, & Zihang Lin Sec 2.2, 2.3 Lecture 4 Notes (by Yikun Zhang)
Further Readings: Paper 1: Unitarily acheivable zero patterns and traces of words in $A$ and $A^*$ ;
Paper 2: Poincare series of some pure and mixed trace algebras of two generic matrices ;
Paper 3: An approximate, multivariate version of Specht's theorem ;
Paper 4: On simultaneous reduction of families of matrices to triangular or diagonal form by unitary congruences
9 Qiwen Zhou & Yue Hu Sec 2.4
10 Mid-term Examination Week (No Class)
11 Sisi Gai & Dan He Sec 2.4
12 Jian Yao & Minghan Dai Sec 2.5
13 Yanran Li & Ruicheng Li Sec 2.6 Python Code: Singular value decomposition for image compression (by Ruicheng Li)
Further Readings: Paper 1: A Singularly Valuable Decomposition: The SVD of a Matrix;
Paper 2: Improving regularized singular value decomposition for collaborative filtering
14 Zihang Lin, Chongshan Xie & Kaixi Wu Sec 2.6, 3.1
15 Zihang Lin, Qiwen Zhou & Yue Hu Sec 2.7, 3.1, 3.2
16 Sisi Gai & Dan He Sec 3.2
17 Jian Yao, Minghan Dai, Kaixi Wu & Chongshan Xie Permutation Matrix, Generalization of Theorem 3.2.3.2 to any field, and Sec 3.3
18 Zihang Lin, Yanran Li & Ruicheng Li Applications of Matrix Analysis in Quantum Mechanics and Neural Network

Homework:

Optional Reading Assignments: A well-known paper describing main results of Linear Algebra without referring to determinants: Down with Determinants!


Mandatory Final Assignment:

After semester-long adventure in Matrix Analysis, we have theoretically scrutinized some fundamental results that facilitate the applications of matrices in practical problems. In this final assignment, you are expected to dabble in real-world applications of Matrix Analysis and experience the power of results that you have learned in lectures.

You are required to select one topic from the following three options, recapitulate the main results of Matrix Analysis that have been applied in the selected topic, and synthesize your understandings and discoveries into a reading report. A self-contained reading report should have at least five sections in order, (i) Abstract (A concise summary of the whole reading report and main findings); (ii) Introduction (A brief overview of the subject or research field that you are going to discuss); (iii) Background (A thorough review of the supporting knowledge for the problem that you are going to investigate; you are only required to review the results that you have learned in this section); (iv) Method and Evaluation (Main section of the reading report. It contains your solutions to the exercises, your methodology to address the problem, and your experimental results); (v) Conclusions (Summarize what you have written in previous sections and discuss your understandings of possible future research directions); (vi) References (Give credit to the papers that you have referred to).

Option 1: The $25,000,000,000 Eigenvector: The Linear Algebra behind Google (A introductory paper discussing Page Rank Algorithm) The problem 1.2.P21 in the textbook and Power Iteration Algorithm in Lecture 2's Notes may give you more insights into the Google matrix. Feel free to resort to an advanced programming platform like Matlab or python when finishing those numerical exercises (Ex.4, 11, 12, 13, 14, etc.). For instance, the functions that you may use in Matlab are eig(), rank(), norm(), eye(), ones(), etc. You don't have to finish all of them if you feel overwhelmed.

Option 2: Make Do with Less: An Introduction to Compressed Sensing (A superficial discussion about Matrix Analysis on compressed sensing) Ask an advanced programming platform for help when you can't obtain the answers to Ex.6(c), 16(c), 17, 19, by hand. The functions that you may use in Matlab are meshgrid(), surf(), fminsearch(), nchoosek(), max(), min(), sum(), etc. Again, discard those exercises that are beyond your level.

Option 3: As some of your classmates have shown, you can search for some applications of main theorems of Matrix Analysis on the Internet, like QR Factorization or Singular Value Decomposition, implement the ideas of those applications via hands-on computation or any computer platform, and compose a complete report. The applications may come from but should not be limited to Computer Science, Physics, Statistics, and so on. Remember to mention the theorems and results in Matrix Analysis that have been used in the report.

The purpose of this assignment is to help you review lecture materials and broaden your horizon in the field of mathematical science. So don't be frustrated if you get stuck in some of the exercises or find it difficult to understand the ideas in any of the above or related papers. Typically, you are not required to submit a long report. An elaborate report with 3-5 pages will guarantee a decent score. Any discussion with the professor, TA, and fellow classmates will be highly encouraged. However, make sure to compose your report by yourself, and any form of plagiarism will not be appreciated.

The reading report can be written by hand, Word, or LaTeX. Here is a LaTeX template (Knitted PDF: file). Your hand-written or printed report should be put into the mailbox of Professor Lixin Yan before the deadline (on the ground floor of the School of Mathematics). You can also submit your final report via email (please send to mcsylx@mail.sysu.edu.cn and yikunzhang@foxmail.com).

Submission Deadline: July 27th, 2018

Some helpful sites or reference books:

Linear Algebra Done Right (A popular textbook for the Advanced Linear Algebra course) (Main page for the book)

Matrix Theory (in Chinese) (This book is designated as the main textbook of the junior optional course "Matrix Analysis" in our department.)

Handbook of Linear Algebra

Topics in Matrix Analysis

The Matrix Cookbook (Facts about matrices)

Maintained by Yikun Zhang