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[parent] elementary matrix (Definition)

Elementary Operations on Matrices

Let $ \mathbb{M}$ be the set of all $ m\times n$ matrices (over some commutative ring $ R$). An operation on $ \mathbb{M}$ is called an elementary row operation if it takes a matrix $ M\in \mathbb{M}$, and does one of the following:

  1. interchanges of two rows of $ M$,
  2. multiply a row of $ M$ by a non-zero element of $ R$,
  3. add a (constant) multiple of a row of $ M$ to another row of $ M$.
An elementary column operation is defined similarly. An operation on $ \mathbb{M}$ is an elementary operation if it is either an elementary row operation or elementary column operation.

For example, if $ M=\begin{pmatrix}a & b \\ c & d \\ e & f \end{pmatrix}$, then the following operations correspond respectively to the three types of elementary row operations described above

  1. $ \begin{pmatrix}a & b \\ e & f \\ c & d \end{pmatrix}$ is obtained by interchanging rows 2 and 3 of $ M$,
  2. $ \begin{pmatrix}a & b \\ rc & rd \\ e & f \end{pmatrix}$ is obtained by multiplying $ r\ne 0$ to the second row of $ M$,
  3. $ \begin{pmatrix}a & b \\ c & d \\ sa+e & sb+f \end{pmatrix}$ is obtained by adding to row 1 multiplied by $ s$ to row 3 of $ M$.

Some immediate observation: elementary operations of type 1 and 3 are always invertible. The inverse of type 1 elementary operation is itself, as interchanging of rows twice gets you back the original matrix. The inverse of type 3 elementary operation is to add the negative of the multiple of the first row to the second row, thus returning the second row back to its original form. Type 2 is invertible provided that the multiplier has an inverse.

Some notation: for each type $ k$ (where $ k=1,2,3$) of elementary operations, let $ E_c^k(A)$ be the set of all matrices obtained from $ A$ via an elementary column operation of type $ k$, and $ E_r^k(A)$ the set of all matrices obtained from $ A$ via an elementary row operation of type $ k$.

Elementary Matrices

Now, assume $ R$ has $ 1$. An $ n\times n$ elementary matrix is a (square) matrix obtained from the identity matrix $ I_n$ by performing an elementary operation. As a result, we have three types of elementary matrices, each corresponding to a type of elementary operations:

  1. transposition matrix $ T_{ij}$: an matrix obtained from $ I_n$ with rows $ i$ and $ j$ switched,
  2. basic diagonal matrix $ D_i(r)$: a diagonal matrix whose entries are $ 1$ except in cell $ (i,i)$, whose entry is a non-zero element $ r$ of $ R$
  3. row replacement matrix $ E_{ij}(s)$: $ I_n + s U_{ij}$, where $ s\in R$ and $ U_{ij}$ is a matrix unit with $ i\ne j$.

For example, among the $ 3\times 3$ matrices, we have

$\displaystyle T_{12} = \begin{pmatrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{... ...quad D_3(r) = \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & r \end{pmatrix},$   and$\displaystyle \quad E_{32}(s) = \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & s & 1 \end{pmatrix}$

For each positive integer $ n$, let $ \mathbb{E}^k(n)$ be the collection of all $ n\times n$ elementary matrices of type $ k$, where $ k=1,2,3$.

Below are some basic properties of elementary matrices:

  • $ T_{ij}=T_{ji}$, and $ T_{ij}^2=I_n$.
  • $ D_i(r)D_i(r^{-1})=I_n$, provided that $ r^{-1}$ exists.
  • $ E_{ij}(s) E_{ij}(-s) = I_n$.
  • $ \det(T_{ij})=-1$, $ \det(D_i(r))=r$, and $ \det(E_{ij}(s))=1$.
  • If $ A$ is an $ m\times n$ matrix, then
    $\displaystyle E_c^k(A)=\lbrace AE \mid E \in \mathbb{E}^k(n) \rbrace$   and$\displaystyle \qquad E_r^k(A)=\lbrace EA \mid E \in \mathbb{E}^k(m) \rbrace.$
  • Every non-singular matrix can be written as a product of elementary matrices. This is the same as saying: given a non-singular matrix, one can perform a finite number of elementary row (column) operations on it to obtain the identity matrix.

Remark. The discussion above pertains to elementary linear algebra. In algebraic K-theory, an elementary matrix is defined only as a row replacement matrix (type 3) above.



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See Also: matrix unit, row reduction

Also defines:  elementary operation, elementary column operation, elementary row operation, basic diagonal matrix, transposition matrix, row replacement matrix

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Cross-references: Algebraic K-theory, linear algebra, column, number, finite, product, non-singular, properties, collection, integer, positive, matrix unit, cell, diagonal matrix, identity matrix, square, negative, inverse, invertible, types, multiple, rows, operation, commutative ring, matrices
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This is version 9 of elementary matrix, born on 2008-10-21, modified 2008-10-22.
Object id is 11195, canonical name is ElementaryMatrix.
Accessed 590 times total.

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AMS MSC: 15-01 (Linear and multilinear algebra; matrix theory :: Instructional exposition )
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