Identity matrix
In mathematics, the identity matrix or identity matrix is a square matrix whose elements are one on the main diagonal and zero everywhere else. The identity matrix is the neutral element in the ring of square matrices with regard to the matrix multiplication . It is symmetrical , selfinverse , idempotent and has maximum rank . The identity matrix is the representation matrix of the identity mapping of a finitedimensional vector space . It is used, among other things, in the definition of the characteristic polynomial of a matrix, orthogonal and unitary matrices, as well as in a number of geometric maps.
definition
If a ring with zero element and one element , then the identity matrix is the square matrix
 .
An identity matrix is therefore a diagonal matrix in which all elements on the main diagonal are the same . In addition to (of identity ), (of unity ) is also used as a spelling . If the dimension is evident from the context, the index is often omitted and only written or .
Examples
If the field of real numbers and denotes and the numbers zero and one , then examples of unit matrices are:
properties
elements
The elements of an identity matrix can be expressed using the Kronecker delta
specify. The identity matrix of size can be just as simple
be noted. The rows and columns of the identity matrix are the canonical unit vectors , and one writes accordingly
 ,
when the unit vectors are column vectors.
neutrality
The following applies to every matrix
 .
Accordingly, the product of any matrix with the identity matrix results in the same matrix again. The set of square matrices, together with the matrix addition and the matrix multiplication, form a (noncommutative) ring . The identity matrix is then the unitary element in this matrix ring , i.e. the neutral element with regard to the matrix multiplication.
Symmetries
The identity matrix is symmetric , that is for their transpose applies
 ,
and selfinverse , that is, for its inverse , also applies
 .
Parameters
The following applies to the determinant of the identity matrix
 ,
which is one of the three defining properties of a determinant. The following applies to the trace of the identity matrix
 .
If it is the ring is , , or , is accordingly obtained . The characteristic polynomial of the identity matrix results as
 .
The only eigenvalue is therefore with multiplicity . In fact applies to all of the module . If a commutative ring , then the rank of the identity matrix is through
given.
Potencies
The identity matrix is idempotent , that is
 ,
and it is the only full ranked matrix with this property. The following applies to the matrix exponential of a real or complex unit matrix
 ,
where is Euler's number .
use
Linear Algebra
The set of regular matrices of size, together with the matrix multiplication, form the general linear group . Then applies to all matrices of this group and their inverses
 .
The center of this group is precisely the multiples (not equal to zero) of the identity matrix. According to the definition, the following applies to an orthogonal matrix
and accordingly for a unitary matrix
 .
These matrices each form subgroups of the corresponding general linear group. The zeroth power of a square matrix is called
set. The identity matrix is used further in defining the characteristic polynomial
a square matrix is used. The identity matrix is the representation matrix of the identity mapping of a finitedimensional vector space .
geometry
In analytical geometry , unit matrices are used, among other things, to define the following mapping matrices:

Point mirroring at the origin of coordinates :

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

Reflection on a straight line through the origin with a unit direction vector :

Reflection on a straight line through the origin (2D) or plane of origin (3D) with a unit normal vector :

Projection onto the complementary space , if a projection matrix is onto a plane or straight line of the origin:
programming
In the numerical software package MATLAB , the unit matrix of the size is generated by the function . In Mathematica , the identity matrix is obtained by .
eye(n)
IdentityMatrix[n]
See also
 Ones matrix , a matrix made up of only ones
 Zero matrix , a matrix that consists of only zeros
 Standard matrix, a matrix that consists of exactly one one and otherwise only zeros
 Permutation matrix , a matrix that is created from a unit matrix by swapping rows or columns
 Elementary matrix, a matrix that differs from an identity matrix only in one position or by exchanging rows
literature
 Siegfried Bosch : Linear Algebra . Springer, 2006, ISBN 3540298843 .
 Karsten Schmidt, Götz Trenkler: Introduction to Modern Matrix Algebra . Springer, 2006, ISBN 3540330089 .
Web links
 Eric W. Weisstein : Identity Matrix . In: MathWorld (English).
 mathcam: Identity matrix . In: PlanetMath . (English)
Individual evidence
 ↑ Christoph W. Überhuber, Stefan Katzenbeisser, Dirk Praetorius: MATLAB 7: An Introduction . Springer, 2007, p. 18 .