Single-entry matrix

A standard matrix standard unit matrix or matrix unit in mathematics, a matrix in which exactly one entry is one and all other entries are zero. Each standard matrix can be represented as a dyadic product of canonical unit vectors. The amount of the standard matrices forms the standard basis for the die space. They are, among others, used to define elementary matrices, which are used in the Gaussian elimination method.

• 4.1 matrix entries
• 4.2 Standard Base
• 4.3 elementary matrices

Definition

Is a ring with zero element and unit element, then the standard matrix is ​​the matrix with the entries

For and. In the standard matrix therefore is the entry at the point equal to one and all other entries equal to zero. A default matrix is also referred to as the default identity matrix or the matrix unit and recorded by occasionally place.

Examples

Is the field of real numbers and name and the numbers zero and one, as are examples of standard matrices of size:

Properties

Representations

Each standard matrix can be described as dyadic product of the two canonical unit vectors and represent, that is,

Where the transposed vector to be. Using the Kronecker delta, a standard matrix can also be

Note.

Symmetry

For the transpose of a matrix standard

Thus, only the standard matrices are symmetric.

Product

For the product of two standard matrices and

Where is the zero matrix of size.

Parameters

For the rank of a standard matrix

For the determinant and the trace of a square matrix standard shall apply mutatis mutandis

The characteristic polynomial of a square matrix standard over a field is given by

In the case, therefore, is the only eigenvalue. For there is additionally the eigenvalue with simple multiplicities and the associated eigenvector.

Use

Matrix entries

With the help of standard dies individual matrix entries can be represented as a trace. Is, then applies

For the product of two matrices and applies accordingly

Standard basis

The amount of the standard matrices over a given body forms the standard basis for the vector space of matrices. Each matrix can therefore by a linear combination of standard dies

With pose. Thus, the four standard dies, and the standard basis of the space of matrices and are obtained for example

Elementary matrices

Standard dies are also used to display the three types of the elementary matrices to the shape

With as the identity matrix and used. By multiplication from the left with a number of such elementary matrix operations, scaling and transpositions are performed on a given matrix. This elementary matrices are used in the description of the Gaussian elimination method for solving systems of linear equations are used.