# Jiyuan Zhang (2018-2021)

## [Principal supervisor till mid 2019 followed by Dr Mario Kieburg]

## Decompositions, invariances and harmonic anlaysis in random matrix theory

### Abstract:

Matrix decompositions are crucial for analysing fixed matrices. Also when the matrix entries become random such decompositions are invaluable. For instance, they can yield factorisations of the measure that simpliifies the analytical problems drastically. In this thesis we use those formulae to explore and link several topics in the common theme of decompositions, invariances and harmonic analysis in random matrix theory. Several applications of those decompositions will be studied.

In this thesis, three types of matrix decompositions are considered: the LU decomposition, the spectral decomposition and the QR decomposition. For the LU decomposition, we will study two topics of parametrising correlation matrices and parametrising compact groups, where the former is an application of random matrix theory in statistics and the latter is motivated by the study of harmonic analysis of random matrices. The LU decomposition is utilised in these two topics due to the convenience it provides in studying positive definite matrices and principal minors.

For the spectral decomposition and the QR decomposition, both of which contain at least one element drawn from a compact matrix group, it is natural to impose invariances to the random matrix models by letting the compact matrix group to be distributed according to the Haar measure. Specially, the spectral decomposition can be applied in the context of harmonic analysis of random matrices, where a link between the historical content of spherical transform and sums and products of random matrices will be shown. We will firstly further develop the theory of sums and products of random matrices using the idea of spherical transforms. Two topics involving the randomised Horn problems and the cyclic Polya ensembles will be studied as the applications.

In the study of the QR decomposition, we will consider random matrix ensembles that have only a one-sided invariance such that the probability density function factorises into a convenient form. We apply this kind of decomposition when we study the Lyapunov exponents of those random matrix ensembles. The QR decomposition reduces those ensembles into a triangular form which is convenient for the computation.

#### Published papers:

Lyapunov exponents for some isotropic random matrix ensembles (with P.J. Forrester), Journal of Statistical Physics **180**, 5581-575 (2020)

Parametrising correlation matrices (with P.J. Forrester), Journal of Multivariate Analysis **178**, 104619 (2020)

Co-rank-1 projections and the randomised Horn problem (with P.J. Forrester), Tunisian Journal of Mathematics **3**, 55-73 (2021)

Co-rank-1 projections and the randomised Horn problem (lead author with M. Kieburg and P.J. Forrester), Letters in Mathematical Physics **111**, 98 (2021)