centralizer of matrix units

Theorem - Let R be a ring with identity 1 and Mn(R) the ring of n×n matrices with entries in R. The centralizerMathworldPlanetmathPlanetmathPlanetmath of the matrix units is the set RId, consisting of all multiples of the identity matrixMathworldPlanetmath.

: 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 Mn(R) as endomorphismsPlanetmathPlanetmath of the module i=1nR. We will denote by {ei} the canonical basis of i=1nR and by Eij the matrix unit whose entry (i,j) is 1.

Let S=[sij]Mn(R) be an element of the centralizer of the matrix units. For all i,j,k we must have

SEijek=EijSek (1)

But a straightforward computation shows that SEijej=Sei and EijSej=sjjei. Since j is arbitrary we see, by equality (1), that all sjj are equal, say sjj=sR.

Hence, Sei=sei, wich means that S=sId. We conclude that S must be a multiple of the identity matrix.

Title centralizer of matrix units
Canonical name CentralizerOfMatrixUnits
Date of creation 2013-03-22 18:39:58
Last modified on 2013-03-22 18:39:58
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 8
Author asteroid (17536)
Entry type Theorem
Classification msc 15A30
Classification msc 16S50