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.

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.)

The ith column of an identity matrix is the unit vector e_i. Using the notation that is sometimes used to concisely describe diagonal matrices, we can write:

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

It can also be written using the Kronecker delta notation:

(I_n)_ij = δ_ij