Editing Identity matrix

{{Short description|Square matrix with ones on the main diagonal and zeros elsewhere}}
{{confuse|matrix of ones|unitary matrix|matrix unit}}
In [[linear algebra]], the '''identity matrix''' of size <math>n</math> is the <math>n\times n</math> [[square matrix]] with [[one]]s on the [[main diagonal]] and [[zero]]s elsewhere. It has unique properties, for example when the identity matrix represents a [[geometric transformation]], the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1.

==Terminology and notation==
The identity matrix is often denoted by <math>I_n</math>, or simply by <math>I</math> if the size is immaterial or can be trivially determined by the context.<ref>{{Cite web|title=Identity matrix: intro to identity matrices (article)| url=https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:properties-of-matrix-multiplication/a/intro-to-identity-matrices | access-date=2020-08-14| website=Khan Academy| language=en}}</ref> 

<math display="block">
I_1 = \begin{bmatrix} 1 \end{bmatrix}
,\ 
I_2 = \begin{bmatrix}
1 & 0 \\
0 & 1 \end{bmatrix}
,\ 
I_3 = \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{bmatrix}
,\ \dots ,\ 
I_n = \begin{bmatrix}
1 & 0 & 0 & \cdots & 0 \\
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1 \end{bmatrix}.
</math>

The term '''unit matrix''' has also been widely used,<ref name=pipes>{{cite book |title=Matrix Methods for Engineering |series=Prentice-Hall International Series in Applied Mathematics |first=Louis Albert |last=Pipes |publisher=Prentice-Hall |year=1963 |page=91 |url=https://books.google.com/books?id=rJNRAAAAMAAJ&pg=PA91 }}</ref><ref>[[Roger Godement]], ''Algebra'', 1968.</ref><ref>[[ISO 80000-2]]:2009.</ref><ref>[[Ken Stroud]], ''Engineering Mathematics'', 2013.</ref> but the term ''identity matrix'' is now standard.<ref>[[ISO 80000-2]]:2019.</ref>  The term ''unit matrix'' is ambiguous, because it is also used for a [[matrix of ones]] and for any [[unit (ring theory)|unit]] of the [[matrix ring|ring of all <math>n\times n</math> matrices]].<ref>{{Cite web| last=Weisstein|first=Eric W.| title=Unit Matrix|url=https://mathworld.wolfram.com/UnitMatrix.html|access-date=2021-05-05| website=mathworld.wolfram.com| language=en}}</ref>

In some fields, such as [[group theory]] or [[quantum mechanics]], the identity matrix is sometimes denoted by a boldface one, <math>\mathbf{1}</math>, or called "id" (short for identity). Less frequently, some mathematics books use <math>U</math> or <math>E</math> to represent the identity matrix, standing for "unit matrix"<ref name=pipes /> and the German word {{lang|de|Einheitsmatrix}} respectively.<ref name=":0">{{Cite web| last=Weisstein|first=Eric W.|title=Identity Matrix | url=https://mathworld.wolfram.com/IdentityMatrix.html|access-date=2020-08-14 | website=mathworld.wolfram.com | language=en}}</ref>

In terms of a notation that is sometimes used to concisely describe [[diagonal matrix|diagonal matrices]], the identity matrix can be written as
<math display=block> I_n = \operatorname{diag}(1, 1, \dots, 1).</math>
The identity matrix can also be written using the [[Kronecker delta]] notation:<ref name=":0" />
<math display=block>(I_n)_{ij} = \delta_{ij}.</math>

==Properties==
When <math>A</math> is an <math>m\times n</math> matrix, it is a property of [[matrix multiplication]] that
<math display=block>I_m A = A I_n = A.</math>
In particular, the identity matrix serves as the [[multiplicative identity]] of the [[matrix ring]] of all <math>n\times n</math> matrices, and as the [[identity element]] of the [[general linear group]] <math>GL(n)</math>, which consists of all [[invertible matrix|invertible]] <math>n\times n</math> matrices under the matrix multiplication operation. In particular, the identity matrix is invertible. It is an [[involutory matrix]], equal to its own inverse. In this group, two square matrices have the identity matrix as their product exactly when they are the inverses of each other.

When <math>n\times n</math> matrices are used to represent [[linear transformation]]s from an <math>n</math>-dimensional vector space to itself, the identity matrix <math>I_n</math> represents the [[identity function]], for whatever [[Basis (linear algebra)|basis]] was used in this representation.

The <math>i</math>th column of an identity matrix is the [[unit vector]] <math>e_i</math>, a vector whose <math>i</math>th entry is 1 and 0 elsewhere. The [[determinant]] of the identity matrix is&nbsp;1, and its [[trace (linear algebra)|trace]] is <math>n</math>.

The identity matrix is the only [[idempotent matrix]] with non-zero determinant. That is, it is the only matrix such that:

# When multiplied by itself, the result is itself
# All of its rows and columns are [[linear independence|linearly independent]].

The [[Square root of a matrix|principal square root]] of an identity matrix is itself, and this is its only [[Positive-definite matrix|positive-definite]] square root. However, every identity matrix with at least two rows and columns has an infinitude of symmetric square roots.<ref>{{cite journal
 | last = Mitchell | first = Douglas W.
 | date = November 2003
 | doi = 10.1017/S0025557200173723
 | issue = 510
 | journal = [[The Mathematical Gazette]]
 | jstor = 3621289
 | pages = 499–500
 | title = 87.57 Using Pythagorean triples to generate square roots of <math>I_2</math>
 | volume = 87| doi-access = free
 }}</ref>

The [[rank (linear algebra)|rank]] of an identity matrix <math>I_n</math> equals the size <math>n</math>, i.e.:
<math display=block>\operatorname{rank}(I_n) = n .</math>

==See also==
* [[Logical matrix|Binary matrix]] (zero-one matrix)
* [[Elementary matrix]]
* [[Exchange matrix]]
* [[Matrix of ones]]
* [[Pauli matrices]] (the identity matrix is the zeroth Pauli matrix)
* [[Householder transformation]] (the Householder matrix is built through the identity matrix)
* [[Square root of a 2 by 2 matrix#Identity matrix|Square root of a 2 by 2 identity matrix]]
* [[Unitary matrix]]
* [[Zero matrix]]

==Notes==
<references />

{{Matrix classes}}

[[Category:Matrices (mathematics)]]
[[Category:1 (number)]]
[[Category:Sparse matrices]]