Elementary matrix

Under an elementary matrix or matrix elimination is understood in linear algebra, a square matrix, which differs either by the change of a single entry or by interchanging two rows of a unitary matrix. Multiplying a matrix on the left by an elementary matrix, this corresponds to an elementary row transformation of the matrix. This matrix transformations include adding the times of a row to another, the interchange of two lines and multiplying a single line with a zero value. The elementary matrices are the basis for the Gauss algorithm. With them, a linear equation system that has been transferred to a matrix, are placed on steps of the form, and then read the solution of the system according to special rules. Elementary matrices can be multiplied by the right to a matrix and then correspond to elementary column operations of.

Types of elementary matrices

There are three types of elementary matrices:

(here, is a body, a unit matrix and a standard matrix (ie, a matrix of zero elements, with the exception that a fuel element is at the location i, j), where i is used as a line index and J a column index of the matrix )

Type 1

This matrix has in its main diagonal elements only one, otherwise only zero elements, with the exception of the point i, j, where the value is, which must be - ie the value may not be in the main diagonal.

This is generated by

For brevity we write

However, note that this is not a standard notation.

Thus performs the following operations:

So this elementary matrix inserts the element at the position ij is the unit matrix.

Examples

Type 2

This matrix corresponds to a unit matrix in which the i-th was swapped with the j- th row (of course). It is on the main diagonal of the points i, i, and j, j weggezählt the unit element (to get zero) and i added at the points i, j and j, the identity element again. In this type, so it is the permutation matrix of a transposition.

The following matrix operations perform this:

For brevity, we define the type 2 as

The operations generally see this:

The following example shows how the ith is swapped with the j- th row:

Example

Is analog

Type 3

The main diagonal of this matrix consists of one element, up to the point i, i where the value is inserted, which must be equal to zero. Outside the main diagonal are all zero elements.

This is achieved by

( At the point i, i is added and subtracted 1. )

For brevity, here is the type 3 as

Be defined. Again, it is not a standard notation.

Operations performed:

This elementary matrix adds in the main diagonal of the matrix is ​​a neutral element.

Examples

Influence of elementary matrices to other matrices

Let A be a matrix and, respectively, and matrices of type 1, type 2 and type 3

Multiplication from the left yields row operations:

  • Multiplying the i-th row of A with the value, wherein the remaining rows remain unchanged ( EPU I)
  • Adds the times of the j- th row of A to the i- th row of A. ( EPU II)
  • Reverses the i-th row of A with the j- th row of A. ( EPU III)

Multiplication from the right yields column operations:

  • Multiplying the i th column with the value of A, the other columns remain unchanged. (ESR I)
  • Adds the times of the i- th column of A to the j -th column of A. ( ESU II) Note the reversed meaning of i and j, in contrast to the line forming.
  • Reverses the i-th column of A with the j -th column of A. ( ESU III)

See also matrix multiplication. These properties are important for solution methods of matrix operations, such as the Gauss-Jordan algorithm.

Hint: To construct the appropriate elementary matrix for one of the above transformations, the corresponding transformation must be applied to the unit matrix. For example, to obtain the elementary matrix, the inverted first and second row of a matrix, the first and second row of the unit matrix are reversed, thereby results.

General characteristics

  • Elementary row operations (or column operations ) are calculated by left multiplication (or right multiplication ) with an elementary matrix.
  • The rank of a matrix is not changed by elementary row or column operations.
  • Is a linear equation system in the form of and may then change the following operations ( by multiplying it by elementary matrices allows ) not to the solution, and are therefore also referred to as elementary transformations (the operations on A and B to be executed at the same time ): Adding the times the value of a row to another row.
  • The interchanging two rows.
  • Multiplying a row with a non-zero value.

Group Theoretical Properties

Whether the group of invertible NxN matrices.

  • Elementary matrices are invertible, and the assignments
  • Every invertible matrix can be written as a product of elementary matrices, ie the elementary matrices generate the group. But suffice even type 1 and type 3 This is the basis also an important application of elementary matrices: To prove a statement for all invertible matrices which satisfy the following two points: It applies to elementary matrices.
  • Does it hold for matrices A and B, it is also true for their product AB.
Identity matrix General linear group Group theory PlanetMath
302802
de