Identity matrix
In linear algebra, the identity matrix of size formula_1 is the formula_2 square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation. The identity matrix is often denoted by formula_3, or simply by formula_4 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 formula_2 matrices. In some fields , such as group theory or quantum mechanics, the identity matrix is sometimes denoted by a boldface one, formula_7, or called "id" (short for identity). Less frequently, some mathematics books use formula_8 or formula_9 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 formula_10 The identity matrix can also be written using the Kronecker delta notation: formula_11 Properties. When formula_12 is an formula_13 matrix, it is a property of matrix multiplication that formula_14 In  particular, the identity matrix serves as the multiplicative identity of the matrix ring of all formula_2 matrices, and as the identity element of the general linear group formula_16, which consists of all invertible formula_2 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 formula_2 matrices are used to represent linear transformations from an formula_1-dimensional vector space to itself, the identity matrix formula_3  represents the identity function, for whatever basis was used in this representation. The formula_21th column of an identity matrix is the unit vector formula_22, a vector whose formula_21th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is formula_1. The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that: 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 formula_3 equals the size formula_1, i.e.: formula_27