matrix unit

A matrix unit is a matrix (over some ring with 1) whose entries are all 0 except in one cell, where it is 1.

For example, among the 3×2 matrices,


are the matrix units.

Let A and B be m×n and p×q matrices over R, and Uij an n×p matrix unit (over R). Then

  1. 1.

    AUij is the m×p matrix whose jth column is the ith column of A, and 0 everywhere else, and

  2. 2.

    UijB is the n×q matrix whose ith row is the jth row of B and 0 everywhere else.

Remarks. Let M=Mm×n(R) be the set of all m by n matrices with entries in a ring R (with 1). Denote Uij the matrix unit in M whose cell (i,j) is 1.

  • M is a (left or right) R-module generated by the m×n matrix units.

  • When m=n, M has the structure of an algebra over R. The matrix units have the following properties:

    1. (a)

      UijUk=δjkUi, and

    2. (b)


    where δij is the Kronecker deltaMathworldPlanetmath and In is the identity matrixMathworldPlanetmath. Note that the Uii form a complete set of pairwise orthogonal idempotents, meaning UiiUii=Uii and UiiUjj=0 if ij.

  • In general, in a matrix ring S (consisting of, say, all n×n matrices), any set of n matrices satisfying the two properties above is called a full set of matrix units of S.

  • For example, if {Uij1i,j2} is the set of 2×2 matrix units over , then for any invertible matrix T, {TUijT-11i,j2} is a full set of matrix units.

  • If we embed R as a subring of Mn(R), then R is the centralizerMathworldPlanetmathPlanetmathPlanetmath of the matrix units of Mn(R), meaning that the only elements in Mn(R) that commute with the matrix units are the elements in R.


  • 1 T. Y. Lam, Lectures on Modules and Rings, Springer, New York, 1998.
Title matrix unit
Canonical name MatrixUnit
Date of creation 2013-03-22 18:30:35
Last modified on 2013-03-22 18:30:35
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 15A30
Classification msc 16S50
Related topic ElementaryMatrix
Defines full set of matrix units