The identity matrix is a the simplest nontrivial diagonal matrix, defined such that

(1)

for all vectors . An identity matrix may be denoted , , (the latter being an abbreviation for the German term "Einheitsmatrix"; Courant and Hilbert 1989, p. 7), or occasionally , with a subscript sometimes used to indicate the dimension of the matrix. Identity matrices are sometimes also known as unit matrices (Akivis and Goldberg 1972, p. 71).

The identity matrix is given explicitly by

(2)

for , ..., , where is the Kronecker delta. Written explicitly,

(3)

The identity matrix is implemented in the Wolfram Language as IdentityMatrix[n].

"Square root of identity" matrices can be defined for by solving

(4)

For , the most general form of the resulting square root matrix is

(5)

giving

(6)

as limiting cases.

"Cube root of identity" matrices can take on even more complicated forms. However, one simple class of such matrices is called k-matrices.