Within the general area of random matrix problems, the PI will especially focus on problems relevant to the following four types of matrices: (i) orthogonal and unitary matrices; (ii) sample correlation matrices; (iii) matrices with matrix-t distribution; (iv) Toeplitz matrices. Based on the PI's and other authors' work on orthogonal matrices, Diaconis has posed an open problem on how a typical random orthogonal matrix can be approximated by a matrix with independent standard normal random variables as entries. This is the motivation to study (i). Part (ii) comes from a statistical hypothesis testing problem when the dimension of a multivariate population distribution and the sample sizes of data from this population are large. By using Principal Component analysis, the maximum eigenvalue of the sample correlation matrix has to be studied. Part (iii) arises from a statistical study on a problem from Image Analysis. The largest entry of matrices with a matrix-t distribution is of central interest. Part (iv) is an unsolved problem in RTM. This type of matrix arises in time series analysis.
The problems studied come from trading markets, engineering and science. The solutions can bring researchers and practitioners from different fields together to exchange ideas: the study helps practitioners by providing new techniques for use and researchers by obtaining motivation and real problems for solution. Matrices are always behind databases. Random matrix theories may give a clean understanding of databases in a certain sense. For example, the largest eigenvalue of a correlation matrix, which is one of the four proposed problems, can tell if multiple quantities depend on each other or not. Further, this work may help graduate students gain a better understanding of this subject.
