SUMMARY
The center of the general linear group GL(n) consists solely of diagonal matrices. The proof involves demonstrating that for any matrix B in the center, the equation BA = AB holds for all matrices A in GL(n). By considering specific simple matrices A, one can effectively show that B must be diagonal. This conclusion is critical for understanding the structure of GL(n) and the role of diagonal matrices within it.
PREREQUISITES
- Understanding of general linear groups, specifically GL(n).
- Familiarity with matrix multiplication and properties of diagonal matrices.
- Knowledge of group theory concepts, particularly the definition of the center of a group.
- Basic linear algebra, including matrix operations and invertibility.
NEXT STEPS
- Study the properties of the center of groups in abstract algebra.
- Learn about the implications of matrix commutativity in linear algebra.
- Explore the structure of GL(n) and its subgroups.
- Investigate the role of diagonalization in linear transformations.
USEFUL FOR
Mathematics students, particularly those studying abstract algebra and linear algebra, as well as educators looking for clear proofs regarding the structure of general linear groups.