Unit Matrix

A unit matrix, in the usage adopted here, is an integer matrix consisting of all 1s (Härdle and Simar 2012, p. 50; Berežný 2014, p. 124, footnote 7; Thulin 2024, Sec. 12.3.1). However, care is needed since the term "unit matrix" is also widely used as a synonym for identity matrix (Akivis and Goldberg 1972, p. 71; Berežný 2014, p. 124, footnote 7). It is also used for an invertible matrix over a commutative ring with a multiplicative identity (MacDuffee 1943, p. 27; Marcus and Minc 1988, p. 69; Marcus and Minc 1992, p. 42).

The unit matrix consisting of all 1s is often denoted , or if . Square unit matrices have determinant 0 for .

For positive integers , , , and , unit matrices satisfy the Kronecker product identity

(1)

In particular, for a positive integer ,

(2)

where denotes the -fold Kronecker product, is the identity matrix, and is the first of the Pauli matrices (Chen and Chen 2025, Eq. 10), where each factor is a unit matrix.

An unit matrix can be generated in the Wolfram Language as ConstantArray[1, m, n].

In the ring-based usage, let be a commutative ring with a multiplicative identity. An square matrix with entries in is a unit matrix when there exists an square matrix such that

(3)

where is the identity matrix.