Identity matrix

The unit matrix or identity matrix in mathematics, a square matrix whose diagonal elements are one and the off-diagonal elements are zero. Is the identity matrix in the ring of square matrices with respect to the neutral element of the matrix multiplication. It is symmetrical, self-inverse, idempotent and has maximal rank. The identity matrix is the matrix representation of the identity map of a finite-dimensional vector space. It is used, among others, in the definition of the characteristic polynomial of a matrix, orthogonal and unitary matrices and in a number of geometric figures.

• 4.1 Linear Algebra
• 4.2 geometry

Definition

Is a ring member with a zero and one element, the unit matrix is ​​the square matrix

An identity matrix is thus a diagonal matrix in which all elements on the main diagonal are equal. As notation is in addition ( of identity) also ( by unit ) are common. If the dimension of the context is clear, is often omitted and only the index or written.

Examples

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

Properties

Components

The components of a unit matrix can be personalized with the Kronecker delta

Specify. The identity matrix of size can be as simple by

Be noted. The rows and columns of the unitary matrix are the canonical unit vector and to write in accordance with

If the unit vectors are column vectors.

Neutrality

For each matrix

Accordingly, the product of an arbitrary matrix of the unit matrix yields again the same matrix. The amount of the square matrices, together with the matrix addition of the matrix multiplication and a ( non-commutative ) ring. The unit matrix is then the identity element in this matrix ring, so with respect to the neutral element of the matrix multiplication.

Symmetries

The identity matrix is symmetric, that is, for its transpose applies

And self-inverse, that is, for its inverse is also true

Parameters

For the determinant of the identity matrix

What is one of the three defining characteristics of a determinant. For the trace of the unit matrix

Is it the ring is, or, is accordingly obtained. The characteristic polynomial of the unit matrix is obtained as

The only eigenvalue is therefore compatible with multiplicity. In fact applies to all of the module. Is a commutative ring, then the rank of the unit matrix by

Given.

Potencies

The unit matrix is idempotent, which means

And it is the only matrix with full rank with this property. Therefore applies to the Matrixexponential a real or complex unitary matrix

The Euler number.

Use

Linear Algebra

The set of regular matrices of size forming the matrix multiplication with the general linear group. Then for all matrices of this group and their inverses

The center of this group are especially the multiple ( non-zero ) of the unit matrix. For an orthogonal matrix by the definition

And according to a unitary matrix

These matrices each form subgroups of the corresponding general linear group. The zeroth power of a square matrix is called

Determined. Further, the unit matrix is used in the definition of the characteristic polynomial

A square matrix used. The identity matrix is the matrix representation of the identity map of a finite-dimensional vector space.

Geometry

In analytical geometry identity matrices are used, inter alia, in the definition of the following imaging matrices:

• Point reflection at the origin:

• Centric stretching with the stretching factor and the origin as center:

• Reflection in a line through the origin with unit direction vector:

• Reflection in a line through the origin ( 2D) or origin plane (3D) with unit normal vector:

• Projection onto the complementary space when a projection matrix on an original level or straight - is:

Programming

In the numerical software package MATLAB the unit matrix of the size generated by the function of eye (s). In Mathematica, the identity matrix is obtained by identity matrix [ n].