centralizer of matrix units
: It is clear that the multiples of the identity matrix commute with all matrix units, and therefore belong to their centralizer. We will now prove the converse.
We will regard the elements of as endomorphisms of the module . We will denote by the canonical basis of and by the matrix unit whose entry is .
Let be an element of the centralizer of the matrix units. For all we must have
But a straightforward computation shows that and . Since is arbitrary we see, by equality (1), that all are equal, say .
Hence, , wich means that . We conclude that must be a multiple of the identity matrix.
|Title||centralizer of matrix units|
|Date of creation||2013-03-22 18:39:58|
|Last modified on||2013-03-22 18:39:58|
|Last modified by||asteroid (17536)|