rank0

Rank 0 is a term used in linear algebra to describe a matrix whose rank is zero. The rank of a matrix A, defined over a field, is the maximum number of linearly independent rows or columns in A. A matrix has rank 0 if and only if every entry is zero, i.e., A is the zero matrix of size m by n. In this case both the row space and the column space consist solely of the zero vector.

Consequences of rank 0 include that the image of the associated linear transformation is the zero vector, and the kernel has full dimension equal to the number of columns n. Thus the rank-nullity relation holds with rank(A) = 0 and nullity(A) = n, so the transformation maps all input vectors to zero.

If the matrix is square, a rank-0 matrix is not invertible and has determinant zero. For a square zero matrix, all eigenvalues are zero. The zero matrix represents the simplest example of a linear transformation with trivial range and maximal kernel.

Examples: a 2×3 zero matrix is rank 0, as is a 3×3 zero matrix. A homogeneous system of linear equations with coefficient matrix A of rank 0 yields Ax = 0 for all x in the domain, since every row is a zero equation.

Generalization: the concept extends to linear transformations between vector spaces. A rank-0 operator maps every vector to the zero vector, and its image is {0} while its kernel is the entire domain.

See also rank, rank-nullity theorem, zero matrix, linear transformation.