The general linear group is a mathematical concept that refers to the set of all invertible matrices over a specified field. In other words, it is the set of all square matrices that have a unique solution when multiplied by any vector.
A diagonal matrix is a square matrix where all the elements outside of the main diagonal (i.e. the diagonal from the top left to the bottom right) are equal to zero. In other words, a diagonal matrix has non-zero elements only on the main diagonal.
The center of the general linear group is the set of all matrices that commute with every other matrix in the group. In other words, if a matrix A is in the center of the general linear group, then for any other matrix B in the group, AB = BA.
The center of the general linear group is important because it contains all the matrices that commute with every other matrix in the group. This property makes it useful in many mathematical calculations and proofs.
The proof involves showing that any matrix in the center of the general linear group can be written as a diagonal matrix. This is done by considering the entries of the matrix and using the commutative property of matrix multiplication. The full proof involves a series of logical steps and mathematical equations.