matrix unit
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
-
1.
is the matrix whose th column is the th column of , and everywhere else, and
-
2.
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.
-
•
When , has the structure of an algebra over . The matrix units have the following properties:
-
(a)
, and
-
(b)
,
where is the Kronecker delta and is the identity matrix. Note that the form a complete set of pairwise orthogonal idempotents, meaning and if .
-
(a)
-
•
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.
-
•
If we embed as a subring of , then is the centralizer of the matrix units of , meaning that the only elements in that commute with the matrix units are the elements in .
References
- 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 |