Identity Matrix

The identity matrix is the 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 size 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].

The matrix square roots of can be obtained by solving

(4)

For , the non-diagonal matrix square roots over the real numbers or complex numbers can be written in the forms

(5)

where and are arbitrary scalars. The diagonal and triangular cases are

(6)

Here may be 0. The signs in the first matrix are independent, while paired signs in the other matrices are chosen together.

Matrices whose cube is the identity can take on even more complicated forms. However, one simple class of such matrices is called k-matrices.