PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (193)
Orphanage
Unclass'd
Unproven (498)
Corrections (22)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
elementary matrix (Definition)

An elementary matrix 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 $ 2\times 2$ matrices,

$\displaystyle \begin{pmatrix} 1 & 0 \ 0 & 0 \end{pmatrix},\quad \begin{pmatri... ...& 0 \ 1 & 0 \end{pmatrix},\quad \begin{pmatrix} 0 & 0 \ 0 & 1 \end{pmatrix}$
are the elementary matrices.

Remarks. Let $ M=M_{m\times n}(R)$ be the set of all $ m$ by $ n$ matrices with entries in a ring $ R$ (with $ 1$). Denote $ E_{ij}$ the elementary matrix whose cell $ (i,j)$ is $ 1$.

  • If $ A\in M$, then $ AE_{ij}$ is the matrix whose columns are all zeros except the $ j$th column, where it is the $ i$th column of $ A$. $ E_{ij}A$ is the matrix whose rows are all zero except the $ i$th row, where it is the $ j$th row of $ A$.
  • $ M$ is a (left or right) $ R$-module generated by the $ m\times n$ elementary matrices.
  • When $ m=n$, $ M$ has the structure of an algebra over $ R$. The elementary matrices have the following properties:
    1. $ E_{ij}E_{k\ell}=\delta_{jk}E_{i\ell}$, and
    2. $ E_{11}+\cdots+E_{nn}=I_n$,
    where $ \delta_{ij}$ is the Kronecker delta and $ I_n$ is the identity matrix. Note that the $ E_{ii}$ form a complete set of pairwise orthogonal idempotents, meaning $ E_{ii}E_{ii}=E_{ii}$ and $ E_{ii}E_{jj}=0$ if $ i\ne j$.
  • In general, in a matrix ring $ S$ (consisting of, say, all $ n\times 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 $ \lbrace E_{ij}\mid 1\le i,j\le 2\rbrace$ is the set of $ 2\times 2$ elementary matrices over $ \mathbb{R}$, then for any invertible matrix $ T$, $ \lbrace TE_{ij}T^{-1}\mid 1\le i,j\le 2\rbrace$ is a full set of matrix units.
  • If we embed $ R$ as a subring of $ M_n(R)$, then $ R$ is the centralizer of the elementary matrices of $ M_n(R)$, meaning that the only elements in $ M_n(R)$ that commute with the elementary matrices are the elements in $ R$.

Bibliography

1
T. Y. Lam, Lectures on Modules and Rings, Springer, New York, 1998.



"elementary matrix" is owned by CWoo.
(view preamble)

View style:

Also defines:  full set of matrix units, matrix unit
Log in to rate this entry.
(view current ratings)

Cross-references: centralizer, subring, invertible, matrix ring, orthogonal idempotents, complete, identity matrix, Kronecker delta, properties, algebra, structure, generated by, right, rows, columns, cell, ring, matrix
There are 5 references to this entry.

This is version 6 of elementary matrix, born on 2007-02-25, modified 2007-03-02.
Object id is 8985, canonical name is ElementaryMatrix.
Accessed 1797 times total.

Classification:
AMS MSC: 16S50 (Associative rings and algebras :: Rings and algebras arising under various constructions :: Endomorphism rings; matrix rings)
 15A30 (Linear and multilinear algebra; matrix theory :: Algebraic systems of matrices)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)