# Center of the general linear group is diagonal matrix proof

• jessicaw
In summary, the problem is to prove that the center of the general linear group consists of diagonal matrices. The attempt at a solution involves multiplying a matrix by its inverse and showing that it remains the same, but this approach has not been successful. An alternative is to consider the equation BA=AB for all matrices A in GL(n) and using specially invented simple matrices to prove the desired result.
jessicaw

## Homework Statement

center of the general linear group is diagonial matrix proof

## The Attempt at a Solution

i write out a n by n matrix and multiply left by a and right by a^-1 and show that it is the same.
I think it can force the matrix to be diagonal but i find it does not work.

Thanks all!

It may be simpler to look at the equation (supposing B is in the center)

BA=AB for all A in GL(n)

and since for all A, then i particular for a bunch of specially invented very simple matrices A.

## What is the general linear group?

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.

## What does it mean for a matrix to be diagonal?

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.

## What is the center of the general linear group?

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.

## Why is the center of the general linear group important?

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.

## How is it proven that the center of the general linear group is a diagonal matrix?

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.

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