A matrix unit is a matrix (over some ring with ) whose entries are all except in one cell, where it is .
For example, among the matrices,
are the matrix units.
Let and be and matrices over , and an matrix unit (over ). Then
is the matrix whose th column is the th column of , and everywhere else, and
is the matrix whose th row is the th row of and everywhere else.
Remarks. Let be the set of all by matrices with entries in a ring (with ). Denote the matrix unit in whose cell is .
is a (left or right) -module generated by the matrix units.
In general, in a matrix ring (consisting of, say, all matrices), any set of matrices satisfying the two properties above is called a full set of matrix units of .
For example, if is the set of matrix units over , then for any invertible matrix , is a full set of matrix units.
- 1 T. Y. Lam, Lectures on Modules and Rings, Springer, New York, 1998.
|Date of creation||2013-03-22 18:30:35|
|Last modified on||2013-03-22 18:30:35|
|Last modified by||CWoo (3771)|
|Defines||full set of matrix units|