nullmatrisen

The nullmatrisen, or the zero matrix, is the m-by-n matrix whose every entry is zero. It is denoted 0_{m×n} or simply 0 when the size is understood. The zero matrix is the additive identity in the set of m×n matrices, since for any A ∈ R^{m×n} one has A + 0_{m×n} = A.

Multiplication by scalars and by other matrices preserves zero: for any scalar c, c·0_{m×n} = 0_{m×n}, and if A is m×n and B is n×p, then A·0_{n×p} = 0_{m×p} and 0_{m×n}·B = 0_{m×p}.

Norms of the zero matrix are zero: the Frobenius norm and all reasonable matrix norms evaluate to 0.

For square matrices of size n×n, the zero matrix has determinant 0, trace 0, and all eigenvalues equal to 0; it is not invertible (except in the degenerate 0×0 case). The rank of 0_{n×n} is 0.

In linear algebra, the zero matrix represents the zero transformation: sending every vector to the origin in the target space. The matrix of the zero linear map from R^n to R^m is precisely 0_{m×n}.

The nullmatrisen is a fundamental object, appearing in solving homogeneous systems of linear equations and as the additive identity in matrix arithmetic. It is unique for each specified size, and the concept is valid over any field or ring in which matrix operations are defined.