Question

Question 0.5. (Centers) Consider the group G is the invertible diagonal matrices. [Hint: each central element must commute with the elements of the form 1Eii where 1 is the identity matrix and Ejj is the matrix with 0's everywhere except a 1 in the ith GLT (R) of invertible n xn matrices. Show that Z(GLn (R)) row and jth column. Why is this element in GL, (R)?]

Answer 1

GLn CR) VA E GL R) AI IA A T A Cainn (Gln (n) AE Z F oi th column. CT+E) A Es) -thm 2 it j a A E GI (R) A az arroan a. O 7 a an A A Eij (E+ Es) an C .. am ain ...Cn anj ani Cith ros) ai A + ani

A + A 2 Gin at atn CaCu C jtheplun A + ajn aj aj1 Car In) AEZ NTwr n EG A A Eij O ho lds fr a 12,i^n ai=ajj for all enhier of A au oft dioal equals dionat enie of A ari Tr A ER T To ou Scala manus with Ser of all 7(GL 1) agonal AtMeR nowzero