Identity matrix

In linear algebra, the identity matrix of size is the 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. The identity matrix is often denoted by , or simply by if the size is immaterial or can be trivially determined by the context. 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 matrices. In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, , or called "id" (short for identity). Less frequently, some mathematics books use or to represent the identity matrix, standing for "unit matrix" and the German word Einheitsmatrix respectively. In terms of a notation that is sometimes used to concisely describe diagonal matrices, the identity matrix can be written as The identity matrix can also be written using the Kronecker delta notation: When is an matrix, it is a property of matrix multiplication that In particular, the identity matrix serves as the multiplicative identity of the matrix ring of all matrices, and as the identity element of the general linear group , which consists of all invertible 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 matrices are used to represent linear transformations from an -dimensional vector space to itself, the identity matrix represents the identity function, for whatever basis was used in this representation. The th column of an identity matrix is the unit vector , a vector whose th entry is 1 and 0 elsewhere.