Identity matrix

Terminology and notation

The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context.

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}.

The term unit matrix has also been widely used, but the term identity matrix is now standard. The term unit matrix is ambiguous, because it is also used for a matrix of ones and for any unit of the ring of all n\times n matrices.

In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, \mathbf{1}, or called "id" (short for identity). Less frequently, some mathematics books use U or E to represent the identity matrix, standing for "unit matrix"

In terms of a notation that is sometimes used to concisely describe diagonal matrices, the identity matrix can be written as I_n = \operatorname{diag}(1, 1, \dots, 1). The identity matrix can also be written using the Kronecker delta notation: (I_n){ij} = \delta{ij}.

Properties

When A is an m\times n matrix, it is a property of matrix multiplication that I_m A = A I_n = A. In particular, the identity matrix serves as the multiplicative identity of the matrix ring of all n\times n matrices, and as the identity element of the general linear group \mathrm{GL}(n), which consists of all invertible n\times n 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 n\times n matrices are used to represent linear transformations from an n-dimensional vector space to itself, the identity matrix I_n represents the identity function, for whatever basis was used in this representation.

The ith column of an identity matrix is the unit vector e_i, a vector whose ith entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is n.

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

1. When multiplied by itself, the result is itself
2. All of its rows and columns are linearly independent.

The principal square root of an identity matrix is itself, and this is its only positive-definite square root. However, every identity matrix with at least two rows and columns has an infinitude of symmetric square roots.{{cite journal

The rank of an identity matrix I_n equals the size n, i.e.: \operatorname{rank}(I_n) = n .

Notes

References

1. "Identity matrix: intro to identity matrices (article)".
2. Pipes, Louis Albert. (1963). "Matrix Methods for Engineering". Prentice-Hall.
3. [[Roger Godement]], ''Algebra'', 1968.
4. [[ISO 80000-2]]:2009.
5. [[Ken Stroud]], ''Engineering Mathematics'', 2013.
6. [[ISO 80000-2]]:2019.
7. Weisstein, Eric W.. "Unit Matrix".
8. Weisstein, Eric W.. "Identity Matrix".