identity matrix
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In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros 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 I_n, or simply by I if the size is immaterial or can be trivially determined by the context. I_1 = \begin 1 \end ,\ I_2 = \begin 1 & 0 \\ 0 & 1 \end ,\ I_3 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ,\ \dots ,\ I_n = \begin 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. 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 In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
or
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, the identity matrix is sometimes denoted by a boldface one, \mathbf, or called "id" (short for identity). Less frequently, some mathematics books use U or E to represent the identity matrix, standing for "unit matrix" and the German word respectively. In terms of a notation that is sometimes used to concisely describe diagonal matrices, the identity matrix can be written as I_n = \operatorname(1, 1, \dots, 1). The identity matrix can also be written using the Kronecker delta notation: (I_n)_ = \delta_.


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 In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
of the matrix ring of all n\times n matrices, and as the
identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
of the
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
GL(n), which consists of all
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
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 In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
e_i, a vector whose ith entry is 1 and 0 elsewhere. The
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
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: # When multiplied by itself, the result is itself # All of its rows and columns are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concep ...
. 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. The rank of an identity matrix I_n equals the size n, i.e.: \operatorname(I_n) = n .


See also

* 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 identity matrix * Unitary matrix * Zero matrix


Notes

{{Matrix classes Matrices (mathematics) 1 (number) Sparse matrices