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Can you help me regarding the Centre of $GL(n,\mathbb{R})$ $?$

It is easy to see that the diagonals are there. what could be the other elements? It may be an useless question but it came to my mind! Any help will be appreciated.

Topology
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If a matrix $A$ is in the center, then it must commute with every invertible matrix. In particular it must commute with elementary matrices which corresponds to the operations on row or columns, depending on the multiplication is made on the left or on the right. In particular if the elementary matrix is $B=$ "multiplication by $a\in\mathbb K$ of the i-th" then you have $AB=BA$; hence multiplying the i-th row by $a\in\mathbb K$ is equivalent to multiplying the i-th column by the same scalar. Hence a matrix in the center must be diagonal.

Viceversa, as you already noticed, a diagonal matrix is in the center. Thus we can conclude that $$ Z(GL_n(\mathbb K))=\operatorname{scal}_n(\mathbb K):=\{a\mathbb I_n\;:\;a\in\mathbb K^{\times}\}\;. $$

Joe
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    Not every diagonal matrix is in the center... –  Jun 16 '14 at 18:45
  • The invertible ones – Joe Jun 16 '14 at 21:39
  • Just pick $a\in\mathbb K\setminus{0}$. Having a look on the considered elementary matrix, it should be clear the role played by $a\in\mathbb K$, if it's zero or not. – Joe Jun 16 '14 at 21:44
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    The $2\times2$ diagonal matrix with diagonal entries 1 and 2 is not in the center of $\text{GL}(2,\mathbb{R})$. –  Jun 17 '14 at 18:41
  • I mean the ones of the type $a\mathbb I_n$, where $a\in\mathbb K$. It results by the proof I wrote – Joe Jun 17 '14 at 22:28
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    Then you are using a non-standard definition of "diagonal matrices". I suggest that you change "diag_n(K)" in your answer to "a I_n for a in K^*", so that others may understand your solution. –  Jun 17 '14 at 22:41
  • See http://en.wikipedia.org/wiki/Diagonal_matrix –  Jun 17 '14 at 22:42
  • The idea of Joe is interesting, but incomplete. Below I suggest how to complete it. Yes, left- and right-multiplications with matrices of type E= 1 0 0 0 1 0 0 0 5 implies that any matrix in the center is diagonal, i.e, elements outside its diagonal are zero. But not every such matrix is in center. Say, A= 2 0 0 0 3 0 0 0 4 is not in center, although commutes with E. A matrix is in center if its diagonal elements are equal (non-zero). Use elementary matrices: F= 1 0 0 0 0 1 0 1 0 To commute with F the matrix A must have equal diagonal elements. – V. Mikaelian Nov 14 '23 at 19:35