The Development of Linear Algebra Research in Hong Kong

by Chi-Kwong Li and Nam-Kiu Tsing, May 14, 1996

Appeared in IMAGE (The Bulletin of the International Linear Algebra Society), Issue 17, Summer of 1996.

A few years ago (about 1989), Bob Thompson suggested us to write an article on the development of linear algebra research in Hong Kong. For various reasons, the project has not been done. We deeply regret that we were not able to do this project earlier so that Bob could see the product before he left us. Actually, Bob had given invaluable help and support for the linear algebra group from Hong Kong. In particular, both authors of this article have benefited a great deal from the generosity of Bob in their careers. In any event, we would like to dedicate this article to our very best friend - Bob Thompson.

Also, in the last few years, there have been a lot of linear algebra activities taking place in Hong Kong. If we do not write up the article now, the project might continue to grow and will be too big for us to handle. This is another motivation for us to do it now.

1. Some Background

It can be said that the Mathematics Department of the University of Hong Kong (HKU) is the cradle of linear algebra research in Hong Kong. Founded in 1911, HKU is the oldest university in the territory, and mathematics is one of the core subjects in its curriculum right from the beginning. But research in mathematics really flourished only after 1948, when Professor Yung-Chow Wong, a geometer, was appointed as Chair of Mathematics. The Department of Mathematics also grew from a teaching staff with only two teachers after the second world war to a faculty of well-trained mathematicians who offer a full range of courses to the students.

Linear algebra has always been regarded as an important subject in the mathematics curriculum of HKU. For mathematics students, the first year core courses are analysis (which include elementary mathematical analysis and multivariable calculus) and linear algebra. A set of lecture notes, which later became a textbook, for the linear algebra course was prepared by Dr. K.T. Leung, who treated the material with mathematical vigor and provided ample examples. This early exposure of students to linear algebra may also have some effects on arousing the interest of students on the subject.

In the seventies, the Mathematics Department of HKU was quite well-developed. Research areas for graduate study included geometry, mathematical analysis, differential equations, number theory, linear algebra, combinatorics, topology, and operations research. In any event, most students were encouraged to pursue their graduate study in other countries such as England, USA and Canada, and many of them became successful mathematicians. But still, for various reasons, there were students who had a strong desire to work with certain faculty members in the department, and decided to study in Hong Kong despite the fact that it would be much harder to get a job at the institutions in Hong Kong after graduation. This remains to be the case even now.

2. The Initiator

Dr. Yik-Hoi Au-Yeung, the initiator of the linear algebra research group in Hong Kong, attended and obtained his first degree from Zhongzhan University in Guangzhou, China, after finishing high school in Hong Kong. Then he spent one and a half year as a graduate student in Fudan University in Shanghai.

In 1962, Dr. Au-Yeung returned to Hong Kong from Shanghai and planned to apply for graduate schools in Australia to study differential equations. The Mathematics Department of HKU was expanding during that period, and its Chairman, Professor Wong, was anxious in recruiting good people. In particular, Dr. Au-Yeung was encouraged to join the department, do his graduate study under Professor Wong, and at the same time do some teaching.

The first research problem that Professor Wong proposed to Dr. Au-Yeung was on eigenvalues of quaternionic matrices. From then on, Dr. Au-Yeung shifted his interest from differential equations to linear algebra. This planted the seeds of future development in linear algebra research in Hong Kong. Dr. Au-Yeung obtained his M.Sc degree in 1966 (thesis title: On the Eigenvalues of Square Quaternion Matrices) and his Ph.D degree in 1970 (thesis title: On Hermitian Functions over Real Numbers, Complex Numbers, or Real Quaternions). He was appointed as assistant lecturer in the department in 1966, and since then has been promoted thrice. He is now a reader in the department.

From the early seventies, Dr. Au-Yeung has been continuously working with a number of graduate students, and maintaining an interest group in linear algebra in the department. For a list of the students of Dr. Au-Yeung, their theses titles, and years of graduation, see Section 5.

3. The Master and His Apprentices

In the seventies and eighties, there were virtually no support from the Hong Kong government or the university for faculty members to do research and advise graduate students. In those days, working with graduate students simply meant asking for more work and burden. Nevertheless, Dr. Au-Yeung devoted his time and efforts to work with graduate students and to develop the linear algebra group.

Dr. Au-Yeung is very liberal in letting his graduate students to choose their research topics, and does not insist them to work on his projects and write joint papers with him. On the other hand, in the same manner as his teacher Professor Wong, he demands research work that is of high quality and quantity from his students. This can be seen from the contents of the theses of some of his students (c.f. Section 5):

And most of these papers were published or accepted for publication before the theses were written.

Dr. Au-Yeung's students are usually trained to use various techniques in a clever way to solve linear algebra problems. As a result, in the early stage of their research careers, many of them have been able to use "low road" (c.f. "High, Low and Quantitative Roads in Linear Algebra" by Robert C. Thompson, in LAA vol.162-164 (1992), pp 23-64) approach to obtain new or reprove existing deep linear algebra results. Also, because of their early training, they would be open to different kinds of approaches (high road, low road, analytic, algebraic, geometric, etc.) to linear algebra problems in their research.

To make sure that the work is of sufficient standard and error-free, Dr Au-Yeung would spend a lot of time to study the papers of his students carefully and give them valuable advices, even when he is not a coauthor of the papers.

Here is a small story that might give some ideas to the readers about Dr. Au-Yeung's attitude towards his graduate students as well as research. In the Spring of 1986, the first author and Tin-Yau Tam were at the final stage of preparing their theses. As usual, everything was in a rush at that point. At that critical moment, Dr. Au-Yeung had to go through an operation of removing his appendix unexpectedly. To avoid delays and to make sure that his students got proper advice in preparing their theses, Dr. Au-Yeung studied the drafts of the theses while he was still in bed in the hospital just after the operation!

To the knowledge of the authors, in the history of HKU there were only two university-wide polls (in 1986 and 1991) for the best teacher. And Dr. Au-Yeung was in both polls voted the best teacher in mathematics by the students.

Dr. Au-Yeung is certainly a highly respected teacher, and is the key figure in the development of linear algebra research in Hong Kong.

4. Other Active Researchers in Hong Kong

The development of linear algebra in Hong Kong has been enhanced by the return of Dr. Raymond Hon-Fu Chan to Hong Kong in 1986. Raymond got his Ph.D degree on numerical linear algebra in New York University (Courant Institute) in 1985 under Professor O.B. Widlund. He joined the Mathematics Department of HKU in 1986, and from 1993 onwards is senior lecturer of the Mathematics Department of the Chinese University of Hong Kong. Raymond's research interests include numerical linear algebra, fast iterative solvers for Toeplitz systems, numerical PDE's and other related topics. He won the Leslie Fox Prize (for best research paper in numerical analysis) in 1989, awarded by the Institute of Mathematics and Its Applications, UK. After returning to Hong Kong, he has been actively involved in various linear algebra activities. He now leads another research group, mainly working on numerical linear algebra and iterative methods, in the Chinese University of Hong Kong. For a list of graduate students he produced please see Section 5.

Another active researcher in linear algebra is Dr. N.N. Chan of the Statistics Department of the Chinese University of Hong Kong.

After spending some years in the USA, the second author rejoined HKU in 1993. He has produced a M.Phil. student.

5. List of Graduates in Linear Algebra

Ph.D. students:

by Dr. Au-Yeung by Dr. Raymond Chan

M.Phil students:

by Dr. Au-Yeung by Dr. Raymond Chan by Dr. Tsing

6. Connections with Linear Algebra Communities Worldwide

Besides producing graduate students, Dr. Au-Yeung and Dr. Raymond Chan are building connections with linear algebra groups in other countries. In particular, Dr. Au-Yeung and his colleagues have organized three mini-conferences on matrix theory in Hong Kong in 1991, 1993 and 1995. Dr. Raymond Chan organized the 1995 Winter School on Iterative Methods in Scientific Computing and Their Applications, and also other conferences and meetings on scientific computing.

There is no doubt that research in linear algebra becomes more and more active. In particular, the linear algebra communities in various Asian countries are growing rapidly. Under this general trend, the Hong Kong linear algebra group will certainly continue to grow healthily, and will contribute to the linear algebra community worldwide.

For any comments and questions, please contact the authors at ckli@math.wm.edu or nktsing@hkuxa.hku.hk