Editing Identity matrix (section)

==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>