RankC

rankC refers to the rank of a matrix when considered over the field of complex numbers, C. The notation is sometimes used to distinguish the complex rank from the rank over the real numbers or other fields. For a matrix with complex entries, rankC(A) is the maximum number of columns (or rows) that are linearly independent when treated as vectors in a complex vector space.

A matrix A over C has rankC(A) equal to the dimension of its column space and also equal to the dimension of its row space. This common value is invariant under row or column operations and applies regardless of whether the matrix is square or rectangular. RankC is consistent with the rank over any larger or different field for the same set of complex entries, and it matches the usual notion of rank used in standard linear algebra.

Common characterizations include: rankC(A) equals the number of nonzero singular values of A; it equals the largest k for which there exists a k×k nonsingular minor of A; equivalently, it is the size of a basis for the column space.

Computation typically proceeds via Gaussian elimination over C or via singular value decomposition, with numerical tolerance used to treat very small values as zero. Applications include solving linear systems, determining the solvability and degrees of freedom, and informing matrix decompositions and control theory analyses in contexts that involve complex data.