In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... For the square matrix section, see square matrix. ... In linear algebra, the main diagonal of a square matrix is the diagonal which runs from the top left corner to the bottom right corner. ...

The important property of I_n is that

AI_n = A   and   I_nB = B

whenever these matrix multiplications are defined. In particular, the identity matrix serves as the unit of the ring of all n-by-n matrices, and as the identity element of the general linear group GL(n) consisting of all invertible n-by-n matrices. (The identity matrix itself is obviously invertible, being its own inverse.) This article gives an overview of the various ways to multiply matrices. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

Where n-by-n matrices are used to represent linear transformations from an n-dimensional vector space to itself, I_n represents the identity function, regardless of the basis. In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...

The ith column of an identity matrix is the unit vector e_i. The unit vectors are also the eigenvectors of the identity matrix, all corresponding to the eigenvalue 1, which is therefore the only eigenvalue and has multiplicity n. It follows that the determinant of the identity matrix is 1 and the trace is n. In mathematics, a unit vector in a normed vector space is a vector (most commonly a spatial vector) whose length is 1. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...

Using the notation that is sometimes used to concisely describe diagonal matrices, we can write: In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ...

I_n = diag(1,1,...,1)

It can also be written using the Kronecker delta notation: In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

(I_n)_ij = δ_ij

External links

• Identity matrix on PlanetMath