In linear algebra, the identity matrix (sometimes ambiguously called a unit matrix) of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context. In some fields, such as quantum mechanics, the identity matrix is denoted by a boldface one, 1; otherwise it is identical to I. Less frequently, some mathematics books use U or E to represent the identity matrix, meaning "unit matrix" and the German word Einheitsmatrix respectively.
When A is m×n, it is a property of matrix multiplication that
In particular, the identity matrix serves as the unit of the ring of all n×n matrices, and as the identity element of the general linear group GL(n) (a group consisting of all invertible n×n matrices). In particular, the identity matrix is invertible--with its inverse being precisely itself.
Using the notation that is sometimes used to concisely describe diagonal matrices, we can write
When the identity matrix is the product of two square matrices, the two matrices are said to be the inverse of each other.
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.