Inlinear algebra, theidentity matrix of size is thesquare matrix with ones on themain diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents ageometric 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.[1]
The termunit matrix has also been widely used,[2][3][4][5] but the termidentity matrix is now standard.[6] The termunit matrix is ambiguous, because it is also used for amatrix of ones and for anyunit of thering of all matrices.[7]
In some fields, such asgroup theory orquantum 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"[2] and the German wordEinheitsmatrix respectively.[8]
In terms of a notation that is sometimes used to concisely describediagonal matrices, the identity matrix can be written asThe identity matrix can also be written using theKronecker delta notation:[8]
When is an matrix, it is a property ofmatrix multiplication thatIn particular, the identity matrix serves as themultiplicative identity of thematrix ring of all matrices, and as theidentity element of thegeneral linear group, which consists of allinvertible matrices under the matrix multiplication operation. In particular, the identity matrix is invertible. It is aninvolutory 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 representlinear transformations from an-dimensional vector space to itself, the identity matrix represents theidentity function, for whateverbasis was used in this representation.
Theth column of an identity matrix is theunit vector, a vector whoseth entry is 1 and 0 elsewhere. Thedeterminant of the identity matrix is 1, and itstrace is.
The identity matrix is the onlyidempotent matrix with non-zero determinant. That is, it is the only matrix such that:
Theprincipal square root of an identity matrix is itself, and this is its onlypositive-definite square root. However, every identity matrix with at least two rows and columns has an infinitude of symmetric square roots.[9]
Therank of an identity matrix equals the size, i.e.:
