Domain Nilpotent Matrices And The Idempotence Keep Function

In resent years，one of the very active research areas in matrix theory is thestudy of preserver problems, including linear preserver problems, additive preserverproblems, multiplicative preserver problems and generic preserver problems. Someresearchers have given a lot of results about it. Problem of preserving idempo-tent relation, preserving nilpotency, and preserving commutativity are some activesubjects of preserver problems.The main results obtained in this thesis are as follows:1. In Chapter1, introduce background and developing state of preserver prob-lems and preserving idempotent, preserving nilpotency and preserving commutativ-ity.2. In Chapter2, maps preserving idempotent relation, preserving nilpotencywere characterized on matrix spaces.3. In Chapter3, maps preserving commutativity relation were characterized ondiferent matrix spaces.