Chen Mufa Talks about the Understanding of Crossover Research
On October 25th, Professor Chen Mufa, academician of CAS was invited into Tianjin University to share his experience and understanding of crossover research with young scholars and students on a main line of exploring the calculation of the eigenvalues and eigenvectors of matrices.
At the beginning of the lecture, Professor Wang Fengyu gave a brief introduction to Professor Chen, and he said that in addition to making a lot of important contributions in the field of probability theory, the research of Professor Chen also involves mathematical physics, differential geometry and random classification of process. In recent years, Professor Chen is very interested in the calculation of the eigenvalues of matrices, and based on his research results in probability theory, he made a wonderful transformation to improve the accuracy and efficiency of the calculation of the eigenvalues and their corresponding eigenvectors. And then, Professor Chen reviewed his mathematical research based on his research experience and background in different period. From the promotion and application of optimum seeking method to production practice in the 1970s, to the research of economic optimization in the late 1980s, and then to the research on numerical mathematics in the past four years, he described himself as a “vagrant”, which is the inspiration of this lecture.
After that, Professor Chen introduced a series of mathematical problems from the approximation calculation of the eigenvector of complex matrix. Behind the seemingly non convergent vector series, there is actually the phenomenon of so-called "wave convergence". Through the tool of "conformal transformation" given in the second part of this lecture, the convergence phenomenon is perfectly explained by Professor Chen. Based on the conclusion that the eigenvalues of real symmetric matrices are all real numbers, Professor Chen moves to look for a larger class of matrices with real spectrum. Through the concept of the symmetrizable matrices, a simple and clear proof reveals a very meaningful result. He shared with the audience the course of research that led to this result. Furthermore, combining with Kolmogorov's circle theorem, the concrete expression of the symmetrizing measure is given.
He then explained the concept of Hermtizable matrices and related theories. In 2018, Professor Chen gave an equivalent criterion ofHermtizable matrix. In particular, he pointed out that for the tridiagonal matrix, this criterion can be further specified, which can be used to describe the birth and death of species, and has a very important application.
He also discussed the algorithm of the eigenvalues and eigenvectorsof matrices. From the perspective of calculation, it is easy to calculate in the case of symmetry, but not in the case of asymmetry. In order to solve these problems, he proposed to use a coupling method to solve this problem, which greatly improved the calculating speed and accuracy. He found that these "improved general algorithms" can be applied to more general matrices, and yield many satisfactory results in many important cases.
Finally, he talked about his personal understanding of the breadth and depth of Scholar. He believed that all disciplines should be a whole and should be crossed and promoted with each other, and as a researcher, we should learn to use "new ideas" to solve the substantive problems in the fields we cross. He thought that when conducting the crossover research, we should pay more attention to the new idea and new method provided by our original research.
At the end of the lecture, Professor Chen patiently answered questions one by one, and gave many suggestions for students on their mathematics studying, and encouraged them to communicate more with others. He earnestly urged everyone to explore the unknown fields from the simple problems. He believed that the most important for research is in facing of challenges and difficulties, we should remain optimistic, frank and honest.