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 , self-inverse , idempotent and has maximum rank . The identity matrix is ​​the representation matrix of the identity mapping of a finite-dimensional 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 ${\displaystyle R}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle I_{n}\in R^{n\times n}}$

${\displaystyle I_{n}={\begin{pmatrix}1&0&\cdots &0\\0&1&\ddots &\vdots \\\vdots &\ddots &\ddots &0\\0&\cdots &0&1\end{pmatrix}}}$.

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 . ${\displaystyle 1}$ ${\displaystyle I_{n}}$ ${\displaystyle E_{n}}$ ${\displaystyle n}$ ${\displaystyle I}$ ${\displaystyle E}$

Examples

If the field of real numbers and denotes and the numbers zero and one , then examples of unit matrices are: ${\displaystyle R}$ ${\displaystyle 0}$ ${\displaystyle 1}$

${\displaystyle I_{1}={\begin{pmatrix}1\end{pmatrix}},\ I_{2}={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\ I_{3}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}},\ I_{4}={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}$

properties

elements

The elements of an identity matrix can be expressed using the Kronecker delta

${\displaystyle \delta _{ij}=\left\{{\begin{matrix}1\quad {\text{falls}}\quad i=j\\0\quad {\text{falls}}\quad i\neq j\end{matrix}}\right.}$

specify. The identity matrix of size can be just as simple ${\displaystyle n\times n}$

${\displaystyle I_{n}=(\delta _{ij})_{i,j\in \{1,\ldots ,n\}}}$

be noted. The rows and columns of the identity matrix are the canonical unit vectors , and one writes accordingly ${\displaystyle e_{1},\ldots ,e_{n}}$

${\displaystyle I_{n}=(e_{1},\ldots ,e_{n})}$,

when the unit vectors are column vectors.

neutrality

The following applies to every matrix ${\displaystyle A\in R^{m\times n}}$

${\displaystyle I_{m}\cdot A=A\cdot I_{n}=A}$.

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 (non-commutative) ring . The identity matrix is ​​then the unitary element in this matrix ring , i.e. the neutral element with regard to the matrix multiplication. ${\displaystyle (R^{n\times n},+,\cdot )}$

Symmetries

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

${\displaystyle (I_{n})^{T}=I_{n}}$,

and self-inverse , that is, for its inverse , also applies

${\displaystyle (I_{n})^{-1}=I_{n}}$.

Parameters

The following applies to the determinant of the identity matrix

${\displaystyle \operatorname {det} (I_{n})=1}$,

which is one of the three defining properties of a determinant. The following applies to the trace of the identity matrix

${\displaystyle \operatorname {spur} (I_{n})=\sum _{i=1}^{n}1}$.

If it is the ring is , , or , is accordingly obtained . The characteristic polynomial of the identity matrix results as ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \operatorname {spur} (I_{n})=n}$

${\displaystyle \chi _{I_{n}}(\lambda )=(\lambda -1)^{n}}$.

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 ${\displaystyle \lambda =1}$ ${\displaystyle n}$ ${\displaystyle I_{n}\cdot x=1\cdot x}$ ${\displaystyle x}$ ${\displaystyle R^{n}}$ ${\displaystyle R}$

${\displaystyle \operatorname {rang} (I_{n})=n}$

given.

Potencies

The identity matrix is idempotent , that is

${\displaystyle I_{n}\cdot I_{n}=I_{n}}$,

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

${\displaystyle \exp(I_{n})=\sum _{k=0}^{\infty }{\frac {(I_{n})^{k}}{k!}}=\sum _{k=0}^{\infty }{\frac {1}{k!}}\cdot I_{n}=e\cdot I_{n}}$,

where is Euler's number . ${\displaystyle e}$

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 ${\displaystyle n\times n}$ ${\displaystyle A}$ ${\displaystyle A^{-1}}$

${\displaystyle A^{-1}\cdot A=A\cdot A^{-1}=I_{n}}$.

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 ${\displaystyle A\in \mathbb {R} ^{n\times n}}$

${\displaystyle A\cdot A^{T}=A^{T}\cdot A=I_{n}}$

and accordingly for a unitary matrix ${\displaystyle A\in \mathbb {C} ^{n\times n}}$

${\displaystyle A\cdot A^{H}=A^{H}\cdot A=I_{n}}$.

These matrices each form subgroups of the corresponding general linear group. The zeroth power of a square matrix is called ${\displaystyle A\in R^{n\times n}}$

${\displaystyle A^{0}=I_{n}}$

set. The identity matrix is ​​used further in defining the characteristic polynomial

${\displaystyle \chi _{A}(\lambda )=\det(A-\lambda I_{n}).}$

a square matrix is ​​used. The identity matrix is ​​the representation matrix of the identity mapping of a finite-dimensional vector space . ${\displaystyle \operatorname {id} \colon V\to V}$ ${\displaystyle V}$

geometry

In analytical geometry , unit matrices are used, among other things, to define the following mapping matrices: ${\displaystyle T}$

• Point mirroring at the origin of coordinates :
  ${\displaystyle T=-I}$
• Centric stretching with the stretching factor and the origin as the center: ${\displaystyle m>0}$
  ${\displaystyle T=m\cdot I}$
• Reflection on a straight line through the origin with a unit direction vector : ${\displaystyle v}$
  ${\displaystyle T=2vv^{T}-I}$
• Reflection on a straight line through the origin (2D) or plane of origin (3D) with a unit normal vector  : ${\displaystyle n}$
  ${\displaystyle T=I-2nn^{T}}$
• Projection onto the complementary space , if a projection matrix is ​​onto a plane or straight line of the origin: ${\displaystyle P}$
  ${\displaystyle T=I-P}$

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 . ${\displaystyle n\times n}$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 3-540-29884-3 .
• Karsten Schmidt, Götz Trenkler: Introduction to Modern Matrix Algebra . Springer, 2006, ISBN 3-540-33008-9 .

Web links

• Eric W. Weisstein : Identity Matrix . In: MathWorld (English).
• mathcam: Identity matrix . In: PlanetMath . (English)

Individual evidence

1. ↑ Christoph W. Überhuber, Stefan Katzenbeisser, Dirk Praetorius: MATLAB 7: An Introduction . Springer, 2007, p. 18 .