elementary matrix (original) (raw)

Elementary Operations on Matrices

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

1. 1. interchanges of two rows of M,
2. 1. multiply a row of M by a non-zero element of R,
3. 1. add a (constant) multiple of a row of M to another row of M.

An elementary column operation is defined similarly. An operation on 𝕄 is an elementary operation if it is either an elementary row operation or elementary column operation.

For example, if M=(abcdef), then the following operations correspond respectively to the three types of elementary row operations described above

1. 1. (abefcd) is obtained by interchanging rows 2 and 3 of M,
2. 1. (abr⁢cr⁢def) is obtained by multiplying r≠0 to the second row of M,
3. 1. (abcds⁢a+es⁢b+f) 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 Eck⁢(A) be the set of all matrices obtained from A via an elementary column operation of type k, and Erk⁢(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×n elementary matrix is a (square) matrix obtained from the identity matrix In by performing an elementary operation. As a result, we have three types of elementary matrices, each corresponding to a type of elementary operations:

1. 1. transposition matrix Ti⁢j: an matrix obtained from In with rows i and j switched,
2. 1. basic diagonal matrix Di⁢(r): a diagonal matrix whose entries are 1 except in cell (i,i), whose entry is a non-zero element r of R
3. 1. row replacement matrix Ei⁢j⁢(s): In+s⁢Ui⁢j, where s∈R and Ui⁢j is a matrix unit with i≠j.

For example, among the 3×3 matrices, we have

For each positive integer n, let 𝔼k⁢(n) be the collection of all n×n elementary matrices of type k, where k=1,2,3.

Below are some basic properties of elementary matrices:

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  Ti⁢j=Tj⁢i, and Ti⁢j2=In.
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  Di⁢(r)⁢Di⁢(r-1)=In, provided that r-1 exists.
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  Ei⁢j⁢(s)⁢Ei⁢j⁢(-s)=In.
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  det⁡(Ti⁢j)=-1, det⁡(Di⁢(r))=r, and det⁡(Ei⁢j⁢(s))=1.
• •
  If A is an m×n matrix, then

  Eck⁢(A)={A⁢E∣E∈𝔼k⁢(n)} and Erk⁢(A)={E⁢A∣E∈𝔼k⁢(m)}.

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  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.

Remarks.

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  One can also define elementary matrix operations on matrices over general rings. However, care must be taken to consider left scalar multiplications and right scalar multiplications as separate operations.
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  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.