On the morning of 28 March 2026, at VNUHCM–University of Science (HCMUS), researcher Lê Quang Trường (class of 2022) successfully defended a doctoral thesis at the institutional level in Algebra and Number Theory. The research is entitled: “Certain matrix decompositions corresponding to given generating sets”.

The thesis was conducted under the academic supervision of Professor Mai Hoàng Biên and Dr Phạm Thị Thu Thủy. The study focuses on investigating matrix decomposition problems within linear groups, particularly describing subgroups generated by specific classes of elements, such as idempotent matrices of index 2, quadratic matrices, matrices of finite order, and commutators.

The scope of the research involves examining the structure of groups such as general linear groups, upper triangular matrix groups, and several classes of extended linear groups. Furthermore, the researcher investigated the representability of elements as products of elements from corresponding generating sets. A significant direction of the thesis is the determination of conditions under which a matrix can be decomposed into prescribed forms, alongside an evaluation of the decomposition length in each specific case.

Regarding the results, the thesis established necessary and sufficient conditions for representing matrices as products of element classes, including idempotent matrices of index 2, quadratic matrices, matrices of finite order, and commutators within specific algebraic classes. Additionally, the research contributes to clarifying the structure of certain subgroups within linear groups and the relationship between decomposition forms and the algebraic properties of matrices.

The findings of the thesis contribute to the supplement and development of research directions concerning linear group structures as well as matrix decomposition problems in non-commutative algebra.

The Examination Board evaluated the thesis as being of high quality, possessing significant scientific merit and professional value within the fields of Algebra and Number Theory. The work was meticulously executed with a systematic approach, yielding numerous original results that do not duplicate previously published studies or research directions. The Board further acknowledged that the researcher possesses strong independent research capabilities, demonstrated through international scientific publications directly related to the thesis content, fully satisfying the requirements for a doctoral degree.