# 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

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

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}$

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

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

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

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

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

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:${\ displaystyle P}$
${\ displaystyle T = IP}$

## 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]

• 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