The zero matrix is the additive identity in . That is, for all it satisfies
There is exactly one zero matrix of any given size m×n having entries in a given ring, so when the context is clear one often refers to the zero matrix. In general the zero element of a ring is unique and typically denoted as 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over any ring.
- Identity matrix, the multiplicative identity for matrices
- Matrix of ones, a matrix where all elements are one
- Single-entry matrix, a matrix where all but one element is zero
- Lang, Serge (1987), Linear Algebra, Undergraduate Texts in Mathematics, Springer, p. 25, ISBN 9780387964126, "We have a zero matrix in which aij = 0 for all i, j. ... We shall write it O."
- Warner, Seth (1990), Modern Algebra, Courier Dover Publications, p. 291, ISBN 9780486663418, "The neutral element for addition is called the zero matrix, for all of its entries are zero."
- Bronson, Richard; Costa, Gabriel B. (2007), Linear Algebra: An Introduction, Academic Press, p. 377, ISBN 9780120887842, "The zero matrix represents the zero transformation 0, having the property 0(v) = 0 for every vector v ∈ V."