The Development of Linear Algebra Research in Hong Kong
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
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
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.
- Part of the M.Phil. thesis of Fuk-Yum Sing constitutes
two journal papers, one of which is on a necessary and
sufficient condition for characterizing diagonal elements
of matrices with prescribed singular values.
This paper was praised highly by Bob Thompson in his review
(Mathematical Reviews, vol. 54 (1977) #12808), who obtained the
same result independently with a different proof just a little earlier.
Material in the Ph.D. thesis of Bit-Shun Tam constitutes five journal papers.
In particular, Tam answered some open problems posed by Hans Schneider and
M. Vidyasagar on cones.
Material in the master thesis of Yiu-Tung Poon constitutes four journal
papers. In one of these papers, Poon gave an ingenious elementary proof
of the convexity of the c-numerical range, which was previously
proved by Roy Westwick using results on Morse theory.
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
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
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
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
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
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
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
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.
by Dr. Au-Yeung
by Dr. Raymond Chan
- Bit-Shun Tam, Some aspects of finite dimensional cones, 1978.
- Nam-Kiu Tsing, A study of generalized numerical ranges, 1983.
- Tin-Yau Tam,
A study of induced operators on symmetry classes of tensors, 1986.
- Chi-Kwong Li, Some results on generalized spectral radii,
numerical radii, and spectral norms, 1986.
- Che-Man Cheng, Some results on eigenvalues, singular values and
orthostochastic matrices, 1991.
- Chi-Fai Chan, Some aspects of generalized numerical ranges and
numerical radii associated with positive semi-definite functions, 1993.
Circulant Preconditioners for Toeplitz Matrices and Their Applications
in Solving Partial Differential Equations, 1992
- K.P. Ng, Some Fast Algorithms in Signal and Image Processing, 1995.
- F.R. Lin, Fast Iterative Methods for Wiener-Hopf Equations, 1995.
by Dr. Au-Yeung
by Dr. Raymond Chan
Fuk-Yum Sing, Some properties on the singular values and diagonal elements
of a matrix, 1977.
Yiu-Tung Poon, Some results on generalized numerical ranges, 1980.
Kam-Chuen Ng, Some properties on doubly-stochastic matrices and the
distribution of density on a numerical range, 1982.
Wai-Yip Man, Some properties of C-numerical ranges and C-numerical
by Dr. Tsing
- K.P. Ng, Fast Iterative Methods for Solving Toeplitz
and Toeplitz-like Systems, 1992.
- C.K. Wong, Block Toeplitz Type Preconditioners for Elliptic
- W.K. Ching, Construction of Preconditioners for Queuing Networks,
H.C. Chan, Iterative Methods for Solving Toeplitz Systems Generated
by Rational Functions, 1995
Hon-Kwok Fung, Some linear preserver problems in system theory, 1995.
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